Plates Shells Lecturenote

8/22/2019 Plates Shells Lecturenote

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2.081J/16.230J Plates and Shells

Professor Tomasz Wierzbicki

Contents

1 Strain-Displacement Relation for Plates 11.1 1-D Strain Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Engineering Strain . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Green-Lagrangian Strain . . . . . . . . . . . . . . . . . . . . . 1

1.2 3-D Strain Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Derivation of Green-Lagrangian Strain Tensor for Plates . . . 11.2.2 Specification of Strain-Displacement Relation for Plates . . . 5

2 Derivation of Constitutive Equations for Plates 102.1 Definitions of Bending Moment and Axial Force . . . . . . . . . . . . 10

2.2 Bending Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Bending Moment . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Bending Energy Density . . . . . . . . . . . . . . . . . . . . . 112.2.3 Total Bending Energy . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Membrane Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Axial Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Membrane Energy Density . . . . . . . . . . . . . . . . . . . 142.3.3 Total Membrane Energy . . . . . . . . . . . . . . . . . . . . . 16

3 Development of Equation of Equilibrium and Boundary ConditionsUsing Variational Approach 17

3.1 Bending Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . 173.1.2 First Variation of the Total Potential Energy . . . . . . . . . 203.1.3 Equilibrium Equation and Boundary Conditions . . . . . . . 243.1.4 Specification of Equation for Rectangular Plate . . . . . . . . 24

3.2 Bending-Membrane Theory of Plates . . . . . . . . . . . . . . . . . . 293.2.1 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . 29

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3.2.2 First Variation of the Total Potential Energy . . . . . . . . . 293.2.3 Equilibrium Equation and Boundary Conditions . . . . . . . 31

4 General Theories of Plate 344.1 Bending Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Derivation of the Plate Bending Equation . . . . . . . . . . . 344.1.2 Reduction to a System of Two Second Order Equations . . . 354.1.3 Exercise 1: Plate Solution . . . . . . . . . . . . . . . . . . . . 364.1.4 Exercise 2: Comparison between Plate and Beam Solution . . 404.1.5 Exercise 3: Finite Difference Solution of the Plate Bending

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Membrane Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Plate Membrane Equation . . . . . . . . . . . . . . . . . . . . 474.2.2 Plate Equation for the Circular Membrane . . . . . . . . . . 48

4.2.3 Example: Approximation Solution for the Clamped Membrane 484.3 Buckling Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 General Equation of Plate Buckling . . . . . . . . . . . . . . 514.3.2 Linearized Buckling Equation of Rectangular Plates . . . . . 524.3.3 Analysis of Rectangular Plates Buckling . . . . . . . . . . . . 544.3.4 Derivation of Raleigh-Ritz Quotient . . . . . . . . . . . . . . 624.3.5 Ultimate Strength of Plates . . . . . . . . . . . . . . . . . . . 654.3.6 Plastic Buckling of Plates . . . . . . . . . . . . . . . . . . . . 714.3.7 Exercise 1: Effect of In-Plane Boundary Conditions, w = 0 . 744.3.8 Exercise 2: Raleigh-Ritz Quotient for Simply Supported Square

Plate under Uniaxial Loading . . . . . . . . . . . . . . . . . . 764.4 Buckling of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.1 Transition from Global and Local Buckling . . . . . . . . . . 774.4.2 Local Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Buckling of Cylindrical Shells 845.1 Governing Equation for Buckling of Cylindrical Shells . . . . . . . . 84

5.1.1 Special Case I: Cylinder under Axial Load P, q = 0 . . . . . 865.1.2 Special Case II: Cylinder under Lateral Pressure . . . . . . . 875.1.3 Special Case III: Hydrostatic Pressure . . . . . . . . . . . . . 885.1.4 Special Case IV: Torsion of a Cylinder . . . . . . . . . . . . . 89

5.2 Derivation of the Linearized Buckling Equation . . . . . . . . . . . . 895.3 Buckling under Axial Compression . . . . . . . . . . . . . . . . . . . 90

5.3.1 Formulation for Buckling Stress and Buckling Mode . . . . . 905.3.2 Buckling Coefficient and Batdorf Parameter . . . . . . . . . . 93

5.4 Buckling under Lateral Pressure . . . . . . . . . . . . . . . . . . . . 965.5 Buckling under Hydrostatic Pressure . . . . . . . . . . . . . . . . . . 995.6 Buckling under Torsion . . . . . . . . . . . . . . . . . . . . . . . . . 1005.7 Influence of Imperfection and Comparison with Experiments . . . . . 102

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1 Strain-Displacement Relation for Plates

1.1 1-D Strain Measure

1.1.1 Engineering Strain

Engineering strain is defined as the relative displacement:

=ds ds0

ds0(1)

where ds0 is the increment of initial lenght and ds is the increment of current length.

1.1.2 Green-Lagrangian Strain

Instead of comparing the length, one can compare the square of lengths:

E=ds2 ds20

2ds20(2)

=ds ds0

ds0

ds + ds02ds0

Where ds ds0, the second term is Eq. (2) tends to unity, and the Green strainmeasure and the engineering strain become identical. Equation (2) can be put intoan equivalnet form:

ds2 ds20 = 2Eds20 (3)which will now be generalized to the 3-D case.

1.2 3-D Strain Measure

1.2.1 Derivation of Green-Lagrangian Strain Tensor for PlatesLet define the following quanties:

a = [ai]: vector of the initial (material) coordinate system

x = [xi]: vector of the current (spatial) coordinate system

u = [ui]: displacement vector

where the index i = 1, 2, 3. The relation between those quantities is:

xi = ai + ui (4)

dxi = dai + dui

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O

a

u

x

du

da dx

O

a

u

x

du

da dx

Now, the squares of the initial and the current length increment can be written

in terms of ai and ui:ds20 = daidajij (5)

ds2 = dxidxj ij (6)

= (dai + dui) (daj + duj) ij

where the Kronecker tensor ij reads:

ij =

1 0 00 1 00 0 1

(7)

The vector u can be considered as a function of:

the initial (material) coordinate system, u (a), which leads to Lagrangiandescripion, or

the current (spatial) coordinates, u (x), which leads to the Eulerian descrip-tion

In structural mechanics, the Lagrangian description is preferable:

ui = ui (ai) (8)

dui =uiak da

k = ui,k dak

duj =ujal

dal = uj ,l dal

Let us calculated the difference in the length square:

ds2 ds20 = (dai + dui) (daj + duj) ij daidajij (9)

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Using Eq. (8) and the definition of ij, the difference in the length square can betransformed into:

ds2 ds20 = (dujdai + duidaj + duiduj) ij (10)= (uj ,l dal dai + ui,k dak daj + ui,k dak uj,l dal) ij

= [uj,l (dajjl) dai + ui,k (daiik) daj + ui,k uj,l (daiik) (dajjl)] ij

= (uj ,i +ui,j +ui,k uj ,k ) daidaj

= 2Eij daidaj

where, by analogy with the 1-D case, the Lagrangian or Green strain tensor Eij isdefined:

Eij =1

2(ui,j +uj ,i +uk,i uk,j ) (11)

In the case of small displacement gradient (uk,i 1), the second nonlinear term

can be neglected leading to the defintion of the infinitesimal strain tensor:

ij =1

2(ui,j +uj,i ) (12)

From the defintion, the strain tensor is symmetric ij = ji, which can be seen byintechanign the indices i for j and j for i. In the moderately large deflection theoryof structures, the nonlinear terms are important. Therefore, Eq. (11) will be usedas a starting point in the development of the general theory of plates.

Components of Green-Lagrangian Strain Tensor Let define the followingrange convention for indices:

Greek letters: ,,... = 1, 2

Roman letters: i,j,... = 1, 2, 3

With this range convention, the Roman letters are also written as:

i = , 3 (13)

j = , 3

The Lagrangian or Green strain tensor can be expressed:

11 12 13

3

21 22 23

31 32 33 3 33

ij

E E E

E EE E EE

E E E E E

= =

# #

# ## #

" " " # " " " " # "

# #

where E is the in-plane component of strain tensor, E3 and E3 are out-of-plane

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shear components of strain tensor, and E33 is the through-thickness component ofstrain tensor. Similarily, displacement vector can be divided into two components:

1

2

3

i

u uu

u vu

u w w

= = =

" " "

where u is the in-plane components of the displacement vector, and u3 = w is theout-of-plane components of the displacement vector and also called as the trans-verse displacement.

Initial Undeformed Configuration

x

z

middle

surface

Deformed Configuration

a

xu

u

3u

Assumptions of the von Karman Theory The von Karman thoery of mod-erately large deflection of plates assumes:

1. The plate is thin. The thickness h is much smaller than the typical platedimension, h L.

2. The magnitude of the transverse deflection is of the same order as the thicknessof plate, |w| = O (h). In practice, the present theory is still a good engineeringapproximation for deflections up to ten plate thickness.

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3. Gradients of in-plane displacements u, are small so that their product orsquare can be neglected.

4. Love-Kirchhoff hypothesis is satisfied. In-plane displacements are a linearfunction of the zcoordiate (3-coordinate).

u = u

z u3, (14)

where u is the displacement of the middle surface, which is independent ofzcoordinate, i.e. u,3 = 0; and u3, is the slope which is negative for the"smiling" beam.

z

5. The out-of-plane displacement is independent of the zcoordiante, i.e. u3,3 =0.

1.2.2 Specification of Strain-Displacement Relation for Plates

In the theory of moderately large deflections, the strain-displacement relation canbe specified for plates.

In-Plane Terms of the Strain Tensors From the general expression, Eq. (11),the 2-D in-plane componets of the strain tensor are:

E =1

2(u, +u, +uk, uk, ) (15)

Here, consider the last, nonlinear term:

uk, uk, = u1, u1, +u2, u2, +u3, u3, (16)

= u, u, +u3, u3,

In the view of the Assumption 3, the first term in the above equation is zero,u, u, ' 0. Therefore, the 2-D in-plane components of strain tensor reads:

E =1

2(u, +u, +w, w, ) (17)

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where w = u3. Introducing Eq. (14) into Eq. (17), i.e. applying Love-Kirchhoffhypothesis, one gets:

E =1

2

(u

z w, ) , + u z w, , +w, w, (18)=

1

2

u, +u

,2 z w, +w, w,

=1

2

u, +u

, z w, + 1

2w, w,

From the definiton of the curvature, one gets:

= w, (19)Now, Eq. (18) can be re-casted in the form:

E = E

+ z (20)

where the strain tensor of the middle surface E is composed of a linear and anonlinear term:

E =1

2

u, +u

,

+1

2w, w, (21)

In the limiting case of small displacements, the second term can be neglected ascompared to the first term. In the classical bending theory of plate, the in-planedisplacements are assumed to be zero u = 0 so that strains are only due to thecurvatue:

E = z (22)

where

=

11 1221 22

=

2w

x2

2w

xy2w

xy

2w

y2

= w, (23)

In the above equation, 11 and 22 are curvatures of the cylindrical bending, and12 is the twist which tells how the slope in the xdirection changes with theydirection:

12 =

y

w

x

for a cylinder

12 0 =

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Interpretation of the linear terms: 12

u, +u

,

Each component can

be expressed in the followings:

11 =12

(u1,1 +u1,1 ) = u1,1 =du1dx

(24)

22 =1

2(u2,2 +u2,2 ) = u2,2 =

du2dy

(25)

12 =1

2(u1,2 +u2,1 ) =

1

2

du1dy

+du2dx

(26)

12|if u2=0 =1

2

du1dy

y

x

11

22

y

x

12

2 0u

1u

y

x

Therefore, 11 and 22 are the tensile strain in the two directions, and 12 is the

change of angles, i.e. shear strain.

Interpretation of the nonlinear term: 12w, w, Let = 1 and = 1.Then, the nonlienar term reads:

1

2w, w,

=1,=1

=1

2

dw

dx

dw

dx=

1

2

dw

dx

2(27)

One can also obtain the same quantity by the defintion of 1-D Green-Lagrangianstrain:

E =ds2 ds20

2ds

2

0

' ds20 + dw

2

ds20

2ds

2

0

=1

2

dw

ds02

=1

2

dw

dx2

(28)

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x

,z w

dx

0ds

dsdw

0ds dx=2 2 2

0ds ds dw= +

Thus, the conclusion is that the nonlinear term 12w, w, represents the change oflength of the plate element due to finite rotations.

Out-Of-Plane Terms of the Strain Tensors Refering to the definition intro-duced in Section 1.2.1, there are three other componets of the strain tensor: E3,

E3 and E33. Using the general expression for the components of the strain tensor,Eq. (11), it can be shown that the application of Assumption 4 and 5 lead to thefollowing expressions:

E3 =1

2(u3, +u,3 +uk,3 uk, ) (29)

=1

2[u3, +u,3 + (u1,3 u1, +u2,3 u2, +u3,3 u3, )]

=1

2[u3, u3, + (u3,1 u1, u3,2 u2, )]

=1

2(u3,1 u1, u3,2 u2, )

=12

w, u,

E3 =1

2(u,3 +u3, +uk, uk,3 ) (30)

=1

2[u,3 +u3, + (u1, u1,3 +u2, u2,3 +u3, u3,3 )]

=1

2[u3, +u3, + (u1, u3,1u2, u3,2 )]

=1

2(u1, u3,1u2, u3,2 )

=12

w, u,

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E33 =1

2(u3,3 +u3,3 +uk,3 uk,3 ) (31)

= u3,3 +1

2h(u1,3 )2 + (u2,3 )2 + (u3,3 )2i

=1

2

h(u1,3 )

2 + (u2,3 )2i

=1

2

h(u3,1 )2 + (u3,2 )2

i=

1

2w, w,

The above are all second order terms which vanish for small deflection theory ofplates. In the theory of moderately larege deflection of plates, the out-of-plate shearstrains as well as the through-thickness strain is not zero. Therefore, an assumption"plane remains plane," expressed by Eq. (14), does not mean that "normal remains

normal." The existance of the out-of-plane shear strain means that lines originallynormal to the middle surface do not remain normal to the deformed plate. However,the incremental work of these strains with the corresponding stresses is negligible:

E33, E33 and E3333, are small (32)

because the corresponding stress 3, 3 and 33 are small as compared to thein-plane stress . One can conclude that the elastic strain energy (and evenplastic dissipation) is well approximated using the plane strain assumption:Z

h

1

2ijijdz '

Zh

1

2dz (33)

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2 Derivation of Constitutive Equations for Plates

2.1 Definitions of Bending Moment and Axial Force

Hooks law in plane stress reads:

=E

1 2 [(1 ) + ] (34)

In terms of components:

xx =E

1 2 (xx + yy ) (35)

yy =E

1 2 (yy + xx)

xy =E

1 +

xy

Here, strain tensor can be obtained from the strain-displacement relations:

=

+ z (36)

Now, define the tensor of bending moment:

M Zh

2

h2

z dz (37)

and the tensor of axial force (membrane force):

N Zh

2

h2

dz (38)

2.2 Bending Energy

2.2.1 Bending Moment

Let us assume that = 0. The bending moment M can be calculated:

M =E

1 2Zh

2

h2

[(1 ) + ] z dz (39)

= E1 2

(1 ) +

Zh2

h2

z dz

+E

1 2 [(1 ) + ]Zh

2

h2

z2 dz

=Eh3

12(1 2) [(1 ) + ]

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Here, we define the bending rigidity of a plate D as follows:

D =Eh3

12(1 2

)

(40)

Now, one gets the moment-curvature relations:

M = D [(1 ) + ] (41)

M =

M11 M12M21 M22

(42)

where M12 = M21 due to symmetry.

M11 = D (11 + 22) (43)

M22 = D (22 + 11)M12 = D (1 ) 12

2.2.2 Bending Energy Density

One -Dimensional Case Here, we use the hat notation for a function of certainargument such as:

M11 = M11 (11) (44)

= D 11

Then, the bending energy density Ub reads :

Ub =

Z110

M11 (11) d11 (45)

= D

Z110

11 d11

=1

2D (11)

2

Ub =1

2M11 11 (46)

11

11M

11d

D

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General Case General definition of the bending energy density reads:

Ub = IM d (47)

22

11

0

Calculate the energy density stored when the curvature reaches a given value .Consider a straight loading path:

= (48)

d = d

0 =

1 =

M

M

M = M () (49)

= M ( )

=

M ()

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where M () is a homogeneous function of degree one.

Ub = IM () d (50)=

Z10

M () d

= M ()

Z10

d

=1

2M ()

=1

2M

Now, the bending energy density reads:

Ub =D2

[(1 ) + ] (51)

=D

2[(1 ) + ]

=D

2

h(1 ) + ()2

iThe bending energy density expressed in terms of components:

Ub =D

2

n(1 )

h(11)

2 + 2 (12)2 + (22)

2i

+ (11 + 22)2o

(52)

=D

2n(1 ) h(11 + 22)2 2 11 22 + 2 (12)2i + (11 + 22)2o

=D

2

nh(11 + 22)

2 2 11 22 + 2 (12)2i

h2 11 22 + 2 (12)2

io=

D

2

n(11 + 22)

2 2 11 22 + 2 (12)2 h2 11 22 + 2 (12)2

io=

D

2

n(11 + 22)

2 + 2 (1 )h11 22 + (12)2

io

Ub =D

2

n(11 + 22)

2 2 (1 )h

11 22 (12)2io

(53)

2.2.3 Total Bending Energy

The total bending energy is the integral of the bending energy density over the areaof plate:

Ub =

ZS

Ub dA (54)

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2.3 Membrane Energy

2.3.1 Axial Force

Assume that = 0. The axial force can be calculated:

N =E

1 2Zh

2

h2

[(1 ) + ] dz (55)

=E

1 2Zh

2

h2

(1 ) +

dz

+E

1 2Zh

2

h2

[(1 ) + ] z dz

=E

1 2 (1 )

+

Zh2

h2

dz

+E

1 2 [(1 ) + ]Zh

2

h2

z dz

=Eh

1 2

(1 ) +

Here, we define the axial rigidity of a plate C as follows:

C =Eh

1 2 (56)

Now, one gets the membrane force-extension relation:

N = Ch

(1 ) + i

(57)

N =

N11 N12N21 N22

(58)

where N12 = N21 due to symmetry.

N11 = C (

11 +

22) (59)

N22 = C (

22 +

11)

N12 = C (1 )

11

2.3.2 Membrane Energy Density

Using the similar definition used in the calculation of the bending energy density,the extension energy (membrane energy) reads:

Um =

IN d

(60)

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Calculate the energy stored when the extension reaches a given value . Considera straight loading path:

=

(61)d =

d

N = N

(62)

= N

= N

where N

is a homogeneous function of degree one.

Um =Z

0N

d (63)

=

Z10

N

d

=1

2N

=1

2N

Now, the extension energy reads:

Um =C

2(1 ) + (64)

=C

2

h(1 ) +

2iThe extension energy expressed in terms of components:

Um =C

2

n(1 )

h(11)

2 + 2 (12)2 + (22)

2i

+ (11 +

22)2o

(65)

=C

2

n(1 )

h(11 +

22)2 2 1122 + 2 (12)2

i+ (11 +

22)2o

=C

2

n(11 +

22)2 2 1122 + 2 (12)2

h2 1122 + 2 (12)2

io

=C

2n

(

11 +

22)2 + 2 (1 ) h1122 + (12)2io

Um =C

2

n(11 +

22)2 2 (1 )

h11

22 (12)2io

(66)

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2.3.3 Total Membrane Energy

The total membrane is the integral of the membrane energy density over the area

of plate::Um =

ZS

Um dS (67)

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3 Development of Equation of Equilibrium and Boundary Condi-tions Using Variational Approach

3.1 Bending Theory of Plates3.1.1 Total Potential Energy

The total potential energy of the system reads:

= Ub Vb (68)

where Ub is the bending energy stored in the plate, and Vb is the work of externalforces.

Bending Energy

Ub =1

2Z

S

M dS (69)

=12

ZS

M w, dS

where the geometrical relation = w, has been used.Work of External Forces

Plate Loading Lateral load:

q(x) = q(x) (70)

This is distributed load measured in [N/m2] or [lb/in2] force per unit area of the

middle surface of the plate.

( )q x( )q x

The distributed load contains concentrated load P as a special case:

P (x0, y0) = P0(x x0) (y y0) (71)

where is the Dirac delta function, [x0, y0] is the coordinate of the application ofthe concentrated force, and P0 is the load intensity.

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x

Dirac -function

0x

NOTE The shearing loads on the lateral surface of ice are normally not consid-ered in the theory of thin plates.

Load Classification

Load applied at the horizontal surfaces.

transverse load

Load applied at the lateral surfaces.

edge force

edge moment

Loads are assumed to be applied to the middle plane of the plateNOTE Other type of loading such as shear or in-plane tension or compression

do not deflect laterally the plate and therefore are not considered in the bendingtheory.

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in-plane tension

in-plane shear

or compression

Potential Energy due to Lateral Load q Lateral (transverse) load doeswork on transverse deflection: Z

Sq w dS (72)

This is also called a work of external forces.

Potential Energy due to Edge Moment The conjugate kinematic variable

associated with the edge moment is the edge rotation dw/dxn.

edge moment

dl

t

n

We apply only the normal bending moment Mnn:

Z

Mnndw

dxndl (73)

where the minus sign is included because positive bending moment results in anegative rotation and negative moment produces positive rotation.

0nnM < 0nnM >

nx0 a

0n

dwdx

> 0n

dwdx

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