Validation of Kirchoff's Plate Theory & its Applicability for Beams

..

1. Validation of Kirchoff’s Plate Theory & its Applicability for Beams b y Abhishek Mondal (12NA30002) Department of Ocean Engineering & Naval Architecture IIT Kharagpur Under the guidance of : Prof. R. Datta Prof. N.Datta

2. Contents Kirchoff’s Plate Theory Problem Formulation Results : Thickness Variation Euler-Bernoulli’s Beam Theory Beam vs Plate Results : Aspect Ration Limitation Future Work References

3. Kirchoff’s Plate Theory It’s an extension of Euler-Bernoulli’s Beam theory to thin plates where aspect ratio is considerable Mid surface plane of a 3-D plate is used to construct a 2-D mathematical model to determine stresses & deformations Shear deformation and rotary inertia are ignored

4. Kirchoff’s Plate Theory The governing differential equation for an isotropic homogeneous plate is : Where h : Plate thickness E : Young’s Modulus q : Load per unit area ρ : Density ν : Poisson’s Ratio

5. Modelling Plate Area : 1 m x 1 m square section E = 210 GPa ρ = 7850 kg /m3 ν = 0.3 Boundary Conditions : i) Fixed Edge : z = 0 & z’ = 0 ii) Free Edge : z’’ = 0 & z’’’ = 0 iii) Sliding Edge : z’ = 0 & z’’’ = 0 iv) Hinged Edge : z = 0 & z’’ = 0

6. Modelling Plate No Left Edge Bottom Edge Right Edge Top Edge Plate Name 1 Hinged Hinged Hinged Hinged SSSS 2 Hinged Free Hinged Free SFSF 3 Clamped Free Clamped Free CFCF 4 Clamped Free Hinged Free CFSF 5 Clamped Free Free Free CFFF 6 Clamped Clamped Clamped Clamped CCCC 7 Free Free Free Free FFFF

7. Plate Contours SSSS SFSF CFCF CFSF CFFF CCCC

8. Results : Fundamental Frequency Plate Name h = 1mm h = 5mm h = 20mm h = 50mm h = 100mm h = 200mm SSSS 4.921 4.918 24.591 24.585 97.746 98.342 239.24 245.854 457.24 491.708 816.47 983.417 SFSF 2.407 2.345 12.007 11.727 47.975 46.907 119.34 117.266 235.14 234.533 447.05 469.065 CFCF 5.5359 5.522 27.673 27.611 110.33 110.44 271.25 276.108 514.62 552.216 873.98 1104.43 CFSF 3.791 3.784 18.952 18.922 75.649 75.688 187.2 189.22 362.48 378.44 651.41 756.881 CFFF 0.8647 0.8646 4.3235 4.323 17.282 17.293 43.073 43.232 85.39 86.465 165.87 172.929 CCCC 8.9803 8.964 44.889 44.821 178.75 179.286 436.25 448.215 809.16 896.43 1312.6 1792.86 FFFF 3.3561 3.355 16.774 16.775 66.754 67.099 164.15 167.747 317.38 335.495 583.59 670.99 Actual Frequency Frequency acoording to Kirchoff’s Theory

9. Results: Thickness Variation

10. Results: Thickness Variation

11. Results: Thickness Variation

12. Thickness Limit (mm) Plate 1 st Mode 2 nd Mode 3 rd Mode 1 st Mode 2 nd Mode 3 rd Mode CFCF 38.6 29.9 24.4 84.5 70.1 58.4 CFFF 88.9 32.3 42 225 96.5 95.2 FFFF 29.8 55.6 52.2 94.2 125.5 116 CCCC 31.5 27.1 22.3 69 53.1 44.3 CFSF 48.5 26.9 22.9 109.7 71.6 59 SSSS 26.5 28.4 21.9 78 66.3 50.6 SFSF 69.4 22.8 22 167 67.9 59.5 1 % Deviation 5 % Deviation

13. Beam Beams are nothing but a special case of plates where aspect ratio (L/B) to is high. It is subjected to transverse load only Moment acts about only one axis Load doesn’t vary in transverse direction

14. Euler–Bernoulli’s Beam Theory Euler-Bernoulli’s beam theory covers the case when deflection is very small compared to the length of the beam. The Euler-Bernoulli equation that describes the relationship between deflection and the applied load is :

15. Euler–Bernoulli’s Beam Theory When there is no external force acting on the beam q = 0 Solving the equation using separation of variable technique w = A 1 cos( β x)+A 2 sin( β x)+A 3 cosh( β x)+A 4 sinh( β x) w here

16. Euler–Bernoulli’s Beam Theory 4 types of beams has been considered viz., Simply Supported Beam [sin( β L )* sinh ( β L ) = 0] Cantilever Beam [ cos ( β L)* cosh ( β L) +1 = 0] Clamped Beam [ cos ( β L)* cosh ( β L) =1 ] Free Beam [ cos ( β L)* cosh ( β L) =1]

17. Simply Supported Beam β 1 L = 3.1416 β 2 L = 6.2832 β 3 L = 9.4248

18. Cantilever Beam β 1 L = 1.875 β 2 L = 4.694 β 3 L = 7.855

19. Clamped Beam β 1 L = 4.73 β 2 L = 7.853 β 3 L = 10.996

20. References Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular platform - L.E . Monterrubio & S . Ilanko The Da Vinci-Euler-Bernoulli Beam Theory – Ballarini & Roberto http://iitg.vlab.co.in /?sub=62&brch=175&sim=1080&cnt=1

21. THANK YOU