Static Analysis of Thin Plates: Bare and Stiffened [308043]

Static Analysis of Thin Plates: Bare and Stiffened

Bhargav Reddy ISANAKA*1,2, Abdul Akbar M2, Vinod KUSHVAHA*1, Biraja Prasad MISHRA2

*Corresponding authors

1[anonimizat], [anonimizat] – 181 221, J&K, India

2[anonimizat] & Science, Angallu, Madanapalle – 517 325, Chittoor, [anonimizat].[anonimizat]*, [anonimizat], [anonimizat]*, [anonimizat]

Abstract: [anonimizat], electronic devices and marine industries etc. [anonimizat]. The present focuses to fill the gap available in the literature to know the plate behaviour of different geometrical shapes before and after adding the stiffener. A [anonimizat], rectangular (with aspect ratio of 1.25, 1.50 and 1.75) and skew square (with 70o, 76.67o and 83.33o skew angles) were considered. Static analysis of considered geometries was carried out by using FEAST (a Finite Element software). [anonimizat]., [anonimizat]. [anonimizat], effect of aspect ratio on rectangular plates and effect of skew angle on skew square plates were given. The displacement of plates reduced significantly after adding the curvilinear stiffener to the bare plates and the stiffener was most effective in the case of circular plate. [anonimizat].

Keywords: Static, plates, geometry, [anonimizat].

1. [anonimizat], civil and electronic device industries. Mechanical, electrical, thermal and tribological properties can be enhanced by using thin plates 1. [anonimizat]-sizes are employed in Microelectromechanical Systems (MEMS) and Nanoelectromechanical Systems (NEMS). [anonimizat], dynamic and buckling behaviour of the plates to explicate their structural performance. The behaviour of plates (bare and stiffened) under static loading condition have been studied for more than five decades 2–6.

[anonimizat] (MFM) was studied by Tamijani and Kapania 7. Their study showed that a curvilinear stiffener performed better compared to a straight stiffener. Liu et al. 8 studied size and surface effects on mechanical behaviour of thin nanoplates by using Isogeometric Analysis (IGA). [anonimizat] a reduction in bending deflection and an increase in buckling loads. Different numerical and analytical methodologies have been used for studying the behaviour of plates (bare and stiffened) under various loading conditions 9–15. Recent studies suggest that curvilinear stiffeners provide more stiffness to the overall plate as compared to the straight stiffeners 16–21. The behaviour of plates for different geometrical shapes by comparing the bare versus stiffened case has not been yet studied. To bridge this gap, different geometrical shapes (circular, square, skew and rectangular) were considered in this study. The dimensions are varied (by maintaining constant plate area) for rectangular and skew geometries by means of varying the aspect ratio (length/breadth) and skew angle respectively. Therefore, in the present study, Finite Element Analysis (FEA) was utilized to study the effect of stiffener for different geometries of the plate under static loading condition.

2. Methodology

Static analysis for the present work was carried out using FEAST (Finite Element Analysis of Structures) software developed by the Indian Space Research Organisation (ISRO) and the Vikram Sarabhai Space Centre (VSSC). Accuracy verification of the finite element model was done by modelling the existing plate geometry with curvilinear stiffeners available in the literature 22 and later the same was compared with the FEAST results. The convergence study plays an important role in the finite element method. Therefore, it must be ensured that an acceptable mesh is used with respect to the shape and size of the elements to achieve the best results. Due to this, in the present work, a convergence study was first carried out for all the considered geometries and then later it was modelled with the converged mesh size.

A total of eight geometrical shapes (refer to Figure 4) were considered in the present study. For the individual comparison of considered geometries, the plate area was kept constant and the origin and shape of the stiffener was same for all the plate geometries. The considered geometries were a circular plate, a square plate, rectangular plates (with aspect ratio 1.25, 1.50, 1.75) and skew square plates (with 70o, 76.67o, 83.33o skew angles). The plates were analysed to understand their behaviour under static loading with and without the curvilinear stiffener. The position of the stiffener is chosen in such a way that it starts from one corner of the plate (except the circular) and ends at the opposite edge of the plate (refer to Figure 4).

In this study, the out-of-plane displacement and Von-Mises stresses were obtained from the static analysis simulations for an applied uniform pressure.

2.1 Validation

The present study was validated by comparing results of a geometrical semi-circular, semi-elliptical plate with curvilinear stiffener (as shown in Figure 1) analysed by using Non-Uniform Rational B-Spline Augmented Finite Element Method (NAFEM) which is proposed by Mishra and Barik 22.

A simply supported semi-circular, semi-elliptical plate having r1 = 12.70 mm and r2= 8.4667 mm, E = 117210 N/mm2, thickness of plate, h = 0.254 mm and poisson ratio, µ = 0.3 with a curvilinear stiffener passing through the points A (-5.9868 mm, -5.9997 mm), B (-0.99776 mm, 1.1459 mm) and C (8.9803 mm, 5.9828 mm) as shown in Figure 1 and was subjected to a uniform pressure of 6.8947 kPa (1.0 Psi) and was analysed using mesh of 32 x 32. The thickness and depth of the stiffener was 0.254 mm and 2.54 mm respectively. Material was identical for both, plate and stiffener.

Figure 1: Semi-circular semi-elliptical plate with curvilinear stiffener passing through the points ABC.

Figure 2: Meshing of semi-circular semi elliptical plate with curvilinear stiffener by

(a) MATLAB coding22 (b) FEAST software

The maximum out-of-plane displacement along minor and major axis were compared. The results obtained from MATLAB and FEAST were in agreement with a maximum error up to 3.08 % along both minor and major axes as shown in Figure 3.

Figure 3: Out of plane displacement along minor and major axes

2.2 Modelling approach

As stated above, one of the methods adopted for comparison of the different geometrical shapes was by maintaining equal area for all plates. Therefore the area of all plates was kept constant. The origin of each geometry was considered at point O. To maintain the constant curvature of the stiffener (but not length), a curve equation was used to calculate the end coordinates of the stiffener (refer to Figure 4). In Figure 4, O and O’are the starting and end point of the curvilinear stiffener and C is the centre of the respective geometry. Geometrical and material properties were taken from the literature 22,23. Geometrical properties of the considered shapes are given in Table 1.

Table 1: Dimensional properties of the considered plates.

Table 2 lists the required properties of plate and stiffener for finite element simulation. For free vibration simulations mass density (ρ) is taken as 4.506 g/cm3. In the present work, the rectangular geometry was analysed for every 0.25 increase in aspect ratio upto 1.75 because for aspect ratio more than 1.75, the considered curvilinear stiffener is not effective. In case of the skew square geometries considered in this study the initial slope of the curvilinear stiffener makes a 70o angle with the horizontal side; due to which the angle considered for skew square geometries is equal to or greater than 70o.

Figure 4: Considered geometries for finite element simulation

Table 2: Properties of a plate and stiffener

3. Results and Discussion

Static analysis for bare and stiffened condition was carried out for all considered geometries namely circular, square, rectangular and skew plates. For all geometrical plates, simply supported boundary condition was adopted and a uniform pressure of 6.8942 kPa was applied. The out-of-plane displacements and Von-Mises stresses were studied and compared.

It was observed that due to the addition of the stiffener to the plate, there was a significant reduction in the displacement compared to the respective bare plate case. Figure 5 gives the comparison of displacement for different geometries (bare vs. stiffened). The displacement decreased four times for the circular stiffened plate compared to the respective bare plate which was maximum among all the considered geometries. Table 3 gives the percentage reduction in displacement for bare and stiffened plates.

In case of bare rectangular plates, the static behaviour was affected by the aspect ratio. From Figures 5 and 6(a), it was observed that the displacement decareased with the increase in aspect ratio for bare plate geometries. The breadth of the plate deceresed with increase in aspect ratio. The reduction in breadth causes plate edges along the length to move closer to the center C of the plate (refer to Figure 4) which decreases the maximum displacement of the bare rectangular plates. However in case of stiffened plates, this behavior is different (refer to Figure 6(a)) due to the combined effect of aspect ratio, stiffener position and geometry. In case of the stiffened plates, with increase in aspect ratio the maximum displacement marginally increases with an increase in the aspect ratio.

The static behaviour of the plate was also affected with increase in skew angle of the bare and siffened plate. With increase in skew angle, the maximum displacement was increased for bare plates (refer to Figure 5) and marginally increased for stiffened plates. Due to the increase in skew angle, the boundaries of the plate move away from the center C of the plate and subsequently showed the increase in out-of-plane deflection (refer to Figure 6(b)).

Among all considered bare plates, the rectangular plate (aspect ratio 1.75) gave the least displacement (0.0046 mm), and among all stiffened plates, skewplate (70o skew angle) produced the minimum displacement (0.0015 mm). Figure 7 shows the displacement contour for all considered geometries of bare and stiffened case.

Figure 5: Comparison of Out-of-plane maximum displacement in bare and stiffened plates

Table 3: Percentage increase in out-of-plane displacement of stiffened plate compared to bare plate due to the stiffener

Figure 6: Effect of (a) aspect ratio (b) skewness on maximum displacement for bare and stiffened plates

Figure 7: Out-of-plane displacement contour of bare and stiffened plates.

Observations were also made based on the obtained individual maximum plate stresses. From Figure 8, the stress is maximum (8.90 MPa) in the skew plate with 70o angle and minimum (5.86 MPa) for the rectangular plate with aspect ratio 1.75 in the bare case. For the stiffened case, the maximum stress (8.38 MPa) is observed for square plate, and minimum (4.21 MPa) for the circular plate.

The stresses decreased in skew plates with increase in skew angle for bare case whereas stresses increased with increase in skew angle for the stiffened case. In the case of rectangular plates, stresses decreased with an increase in aspect ratio for the bare case. But for stiffened rectangular plates there was no clear increasing or decreasing trend. Due to the curvilinear stiffener, the maximum stress occurred at the bottom left corner of the rectangular and square plates (in the path and the vicinity of the stiffener, refer to Figure 9). Due to which, in the case of square and rectangular plates with aspect ratios 1.25 and 1.75 the magnitude of stress was even higher as compared to the respective bare plate cases. Figure 10 shows the stress contour of all considered geometries.

Figure 8: Comparison of maximum von-mises stresses in bare and stiffened plates

Figure 9: Maximum stress hotspot for stiffened square plate

Figure 10: Stress contour of bare and stiffened plates.

4. Conclusions

Static analysis of thin plates (bare vs. stiffened) with simply supported boundary conditions, isotropic material properties and curvilinear stiffener with a constant curvature for all considered geometries gave some of the important conclusions based on the plate behaviour. From the present study, the following conclusions were obtained.

Displacement of plates reduced significantly after adding stiffener to the bare plates.

Among all the bare plates, displacement is minimum for the rectangular plate (with aspect ratio 1.75) and maximum for the circular plate.

In rectangular plates, displacement is reduced with an increase in aspect ratio for the bare case but increased for the stiffened case.

In bare plates, (in skew ones) displacement increased with the increase in skew angle but only a marginal increase was observed in the stiffened case.

Among the considered geometries the stiffener was most effective in the case of the circular plate (displacement reduced by four times compared to bare case).

The stress is maximum (8.90 MPa) in the skew plate with 70o angle and minimum (5.86 MPa) for a rectangular plate with aspect ratio 1.75 in bare case. For the stiffened case, the maximum stress (8.38 MPa) is observed for the square plate and minimum (4.21 MPa) for the circular plate.

With the increase in skew angle, stresses decreased for the bare case whereas stresses increased with increase in skew angle for the stiffened case.

The maximum stress occurred at the bottom left corner of the rectangular and square plates (in the path and the vicinity of the stiffener) due to the curvilinear stiffener. Due to this reason, in case of square and rectangular plates with aspect ratios 1.25 and 1.75, the magnitude of stress was even higher as compared to the respective bare plate cases.

5. Acknowledgement

Authors would like to thank Mr. Aniket Bhelsaikar, CEA support engineer of SVR Info Tech, for his support to understand FEAST and he provided insight and expertise that greatly assisted the research. We would also like to thank ISRO/VSSC for providing the software tool for the present study.

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