Mini project For M.tech Structural Engineering Deflection of Simply supported thin Rectangular plates

1. Deflection of Simply
Supported Thin Rectangular
Plates

2. Contents
1. Introduction of Plates
2. Navier’s Method (Double Series
Solution)
3. Levy’s Method (Single Series
Solution)
4. Comparision of Navier’s and Levy’s
Method
5. Conclusion

3. ABSTRACT
When a plate introduced to external load
will have a deformation out of its own
plane.
comparison of simply supported plate
deformation using Navier’s solution and
Levy’s Solution

4. Plates
• A plate resists transverse loads by
means of bending, exclusively.
• Plates may be classified into three
groups according to the ratio a=h,
where a is a typical dimension of a plate
in a plane and h is a plate thickness.

5. Classification of Plates
• Thick Plates - having ratios a/h 8 . . .10
• Membranes - with ratios a/h 80 . . . 100
• Thin Plates - Intermediate type of plate,
a/h with 8 . . . 10 a/h 80 . . . 100.Depending
on the value of the a/h.

6. Thin Plates
• Thin plates are initially flat structural
members bounded by two parallel
planes, called faces, and a cylindrical
surface, called an edge or boundary. The
generators of the cylindrical surface are
perpendicular to the plane faces. The
distance between the plane faces is
called the thickness (h) of the plate

7. Behavior of Plates
• The behavior of plates is similar to that of
beams. They both carry transverse loads by
bending action.
– Plates carry transverse loads by bending and shear
just like beams, but they have some peculiarities
– We will focus on isotropic homogenous plates.
x
y
z
Simply supported edges

8. Behavior of Plates
x
y
z
Simply supported edges
q
 Plates undergo bending
which can be represented
by the deflection (w) of the
middle plane of the plate
u
vw
w(x,y) w(x+dx,y)
w(x+dx,y+dy)
w(x,y+dy)
w/x
w/y
 The middle plane of the plate undergoes deflections w(x,y). The top
and bottom surfaces of the plate undergo deformations almost like a
rigid body along with the middle surface.

9. Bending of a plate
• A plate has a wide cross section
- top and bottom edge of a cross section remain
straight y-parallel axis when Mx is applied
• When a plate is bent to a cylindrical surface, only
Mx acts:
)1(12)1(
stiffnessBending
000
2
3
2
2
2
2
2
2
2
















EtEI
D
y
w
y
w
x
w
M xyx

10. Assumptions :
Linear thin plate theory is developed under
the following assumptions:
• The plate thickness, t, is much smaller
than the plates dimensions in the x-y-
plane.
• Small deformations (less than 0.3 times
the thickness).
• The plates material is linear elastic,
homogeneous, and isotropic.

11. • Example :-
• A rectangular plate of sides a and b is
simply supported on all edges and
subjected to a uniform pressure P as
shown in Fig. To Determine the
maximum deflection using Navier’s
Method and Levy’s Method.

13. • This solution method can be used for a
plate that is pinned along all edges. The
method gives the displacement function
from an arbitrary distributed load.

14. • this method can be used on a plate with
pinned edges.
• The boundary conditions can then be
formulated as:
w = 0 and w,nn = 0 (or Mn = 0) on all edges.

15. Deflection(w) and Moments(M)

16. Transverse Shear Forces(Q) &
Effective Shear Forces(V)

17. The method consists of a Fourier series
development of an arbitrary load P Then
displacement:-

18. • When the displacement w(x,y) is known,
inner moment, shear forces and stresses
can be calculated. This calculation
method has quick convergence for load
distributions that covers the entire plate.
• And the method is well suited for solution
on computers.

19. • Another approach was proposed by Levy
in 1899. In this case we start with an
assumed form of the displacement and
try to fit the parameters so that the
governing equation and the boundary
conditions are satisfied.
Contd.

20. • This method is used for plates with at
least two pinned edges. The other two
can have arbitrary boundary conditions.
• Presently we are considering simply
supported conditions on four sides.
Contd.

21. Contd.

22. Contd.
The general steps in Levy's solution is therefore:
• Calculate the coefficients in the load Fourier
series.
• Establish the particular solution for all m.
• Superposition of the homogeneous and
particular solution.
• The boundary conditions on all the sides with
arbitrary support to determine the constants
Am, Bm, Cm and Dm.

23. Contd.
The displacement function must also satisfy the
differential equation for the plate, and the
boundary conditions at y = 0 and y = b. To
establish a solution, it is practical to divide the
displacement function into one homogeneous
solution and one particular solution:
w(x,y) = wH(x,y)+wP(x,y)

24. • After some calculations, you will end up with
an expression for the displacement:
• The expression contains four unkown
constants for each m, Am, Bm, Cm and Dm. The
boundary conditions on the two sides with
arbitrary support is used to determine these
constants.

25. • The method has quick convergence, and
few parts are needed to establish a good
expression for w(x,y). If moment and
stresses are to be determined with a
minimal error, more parts must be taken
into consideration.
Contd.

26. • Consider a square plate under uniformly
distributed load then the results are
obtained from the following sheet as

27. Results by Navier’s Method – Table 1
m, n Values
W max
Ʃm, n Values upto
At (a/2, a/2)
1 4.8220 4.8220
3 0.0298 4.8518
5 0.0112 4.8630
7 0.0003 4.8633
13 0.0003 4.8648
15 0.0000 4.8648
17 0.0001 4.8649
19 0.0000 4.8649

28. Results by Navier’s Method – Table 1
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Variation of Wmax with m&n values
m and n values X axis Deflection in mm in Y axis

29. Note :
µ is taken to be 0.3.
(x,y) locations are chosen correspond to maximum
deflection or stress resultant.
• It should be noted from this table that the
rate of convergence of the series is very
fast for w.
Results by Navier’s Method

30. • The accuracy of the one-term
approximation for Wmax is very good
(error with respect to the converged
solution is approximately 2.4).
Results by Navier’s Method

31. Results by Levy’s Method – Table 2
m, n Values
W max
Ʃm, n Values upto
At (a/2, a/2)
1 4.8118 4.8118
3 0.0527 4.8645
5 0.0000 4.8645
7 0.0000 4.8645
9 0.0000 4.8645
11 0.0000 4.8645
13 0.0000 4.8645
15 0.0000 4.8645
17 0.0000 4.8645
19 0.0000 4.8645

32. Results by Levy’s Method – Table 2
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Variation of Wmax with m&n values
Values of m & n X axis Deflection in mm Y axis

33. Results by Levy’s Method
Note :
µ is taken to be 0.3.
(x,y) locations are chosen correspond to
maximum deflection or stress resultant.

34. Results by Levy’s Method
Conclusion:
• I want to conclude that a comparision of
the above results with those of Table 1
reveals the superior convergence of
Levy’s Method compares to that of
Navier’s Method ; accurate estimates of
both deflections can be obtained hereby
considering just the first few terms.

35. • Analysis of Plates by T.K Varadan &
K.Bhaskar.
• Class Note Book
• Guidance from V.Ramesh Sir.

36. Thank You