Deflection of Simply Supported Thin Rectangular

Plates

Contents1. Introduction of Plates2. Navier’s Method (Double

Series Solution)3. Levy’s Method (Single Series

Solution)4. Comparision of Navier’s and

Levy’s Method5. Conclusion

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

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.

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.

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

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

yzSimply supported edges

Sim

ply su

ppor

ted

edge

s

Behavior of Plates

x

yzSimply supported edges

Sim

ply su

ppor

ted

edge

s

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.

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( stiffness Bending

0 0 0

2

3

2

2

2

2

2

2

2

EtEID

yw

yw

xwM xyx

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.

• 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.

• 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.

1 . NAVIER’S METHOD (DOUBLE SERIES SOLUTION)

• 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.

1 . NAVIER’S METHOD (DOUBLE SERIES SOLUTION)

Deflection(w) and Moments(M)

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

1 . NAVIER’S METHOD (DOUBLE SERIES SOLUTION)

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

1 . NAVIER’S METHOD (DOUBLE SERIES SOLUTION)

• 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.

• 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.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

Contd.

• 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.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

Contd.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

Contd.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

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.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

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)

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

• 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.

• 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.

2. LEVY’S METHOD(SINGLESERIES SOLUTION)

Contd.

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

COMPARISION OF TWO METHODS

Results by Navier’s Method – Table 1

COMPARISION OF TWO METHODS

m, n ValuesW max

Ʃm, n Values uptoAt (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

Results by Navier’s Method – Table 1

COMPARISION OF TWO METHODS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

10

20

30

40

50

60

Variation of Wmax with m&n values

m and n values X axis Deflection in mm in Y axis

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

COMPARISION OF TWO METHODS

• 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

COMPARISION OF TWO METHODS

Results by Levy’s Method – Table 2

COMPARISION OF TWO METHODS

m, n ValuesW max

Ʃm, n Values uptoAt (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

Results by Levy’s Method – Table 2

COMPARISION OF TWO METHODS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

10

20

30

40

50

60

Variation of Wmax with m&n values

Values of m & n X axis Deflection in mm Y axis

Results by Levy’s Method

COMPARISION OF TWO METHODS

Note :µ is taken to be 0.3.(x,y) locations are chosen correspond to maximum deflection or stress resultant.

Results by Levy’s Method

COMPARISION OF TWO METHODS

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.

BIBILOGRAPHY

• Analysis of Plates by T.K Varadan & K.Bhaskar.

• Class Note Book• Guidance from V.Ramesh Sir.

Thank You