This document summarizes research validating Kirchhoff's plate theory and investigating its applicability for beams. The document examines Kirchhoff's plate theory, establishes models of various plate configurations, analyzes results on fundamental frequency and thickness variation limitations. It also compares plate and beam theories, examining Euler-Bernoulli beam theory and the relationship between beams and plates as aspect ratio increases. The research finds thickness limitations for Kirchhoff's plate theory and shows convergence of plate and beam fundamental frequencies as aspect ratio increases.

1. Validation of Kirchoff’s Plate
Theory & its Applicability for
Beams
by
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
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
FrequencyRatio
Thickness (m)
Deviation from Kirchoff's Plate Theory (1st Mode)
CFCF CFFF FFFF CCCC CFSF SSSS SFSF

10. Results: Thickness Variation
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
FrequencyRatio
Thickness (m)
Deviation from Kirchoff's Plate Theory (2nd Mode)
CFCF CFFF FFFF CCCC CFSF SSSS SFSF

11. Results: Thickness Variation
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
FrequencyRatio
Thickness(m)
Deviation from Kirchoff's Plate Theory (3rd Mode)
CFCF CFFF FFFF CCCC CFSF SSSS SFSF

12. Thickness Limit (mm)
Plate 1st
Mode
2nd
Mode
3rd
Mode
1st
Mode
2nd
Mode
3rd
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 = A1cos(βx)+A2sin(βx)+A3cosh(βx)+A4sinh(βx)
where

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

17. Simply Supported Beam
3
3.1
3.2
3.3
3.4
1 2 3 4 5 6 7 8 9 10
(βL)-Value
Aspect Ratio (L/B)
Plate
Beam
0
2
4
6
8
10
1 2 3 4 5 6 7 8 9 10
(βL)-Value
Aspect Ratio (L/B)
Plate
Beam
0
2
4
6
8
1 2 3 4 5 6 7 8 9 10
(βL)-Value
Aspect Ratio (L/B)
Plate
Beam
β1L = 3.1416
β2L = 6.2832
β3L = 9.4248

18. Cantilever Beam
0
2
4
6
8
10
1 2 3 4 5 6 7 8 9 10 11
(βL)-value
Aspect Ratio (L/B)
Plate
Beam
0
5
10
15
1 2 3 4 5 6 7 8 9 10 11
(βL)-value
Aspect Ratio (L/B)
Plate
Beam
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 11
(βL)-value
Aspect Ratio (L/B)
Plate
Beam
β1L = 1.875
β2L = 4.694
β3L = 7.855

19. Clamped Beam
0
2
4
6
8
10
1 2 3 4 5 6 7 8 9 10 11
Series1
Series2
β1L = 4.73
β2L = 7.853
β3L = 10.996
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11
(βL)-value
Aspect Ratio (L/B)
Plate
Beam
4
4.5
5
5.5
6
1 2 3 4 5 6 7 8 9 10 11
(βL)-value
Aspect Ratio (L/B)
Plate
Beam

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