section 13 plate elements.ppt

Section 13: PLATE BENDING ELEMENTS

Washkewicz College of Engineering

Plate Theory

A plate is a structural element whose mid surface lies in a flat plane. The dimension in the direction normal to the plane is referred to as the thickness of the plate. A plate is characterized by a small thickness relative to the other dimensions of the plate.

Plates may be classified by the relative thickness of the structural element. We can speak of:

1. Relatively thick plates with small displacements

2. Relatively thin plates with small displacements

3. Very thin plates with large deflections

4. Extremely thin plates (membranes) with large or small deflections

If a plate is curved as opposed to flat, we refer to these structural elements as shells. We start the discussion with classical thin plate theory attributed to Kirchoff

1



Plate Geometry and Deformation

The basic derivation of the field equations for thin plates starts with the geometry shown in the figure below:

The plate surfaces are at

and the mid-surface is at0z

2tz

2



Note that

and if t is more than about one-tenth the span of the plate, then transverse shear deformations must be accounted for and the plate is said to be “thick.” Also note that the transverse deflection w is much smaller than the thickness t, i.e.,

These are things that should be checked when running a finite element analysis with thin plate elements.

ctbt

1tw

Analogous to beams there are several types of plate theories associated with geometric assumptions.

3



Kirchhoff Assumptions and Theoretical Formulation

Consider a differential slice from a plate with cutting planes perpendicular to the x-axis as shown below. The load q from the previous figure causes the plate to deform laterally in the positive z direction. The deflection at any point P in the plate is functionally dependent on its position relative to the x-y plane, i.e.,

and the plate does not stretch in the z-direction. The line a-b drawn perpendicular to the mid-surface before loading remains perpendicular to the mid-surface after loading.

yxww ,

4



This is consistent with the Kirchoff assumptions which are stated as follows:

1. Straight lines normal to the mid surface remain normal. This implies that

However

that is right angles in the plane of the plate do not necessarily remain right angles after the plate is loaded.

2. Thickness changes are neglected and lines normal to the mid surface do not undergo extension. This implies

Normal stresses are considered negligible.

3. Membrane forces are neglected. Plane stress solutions are superimposed later in the plane of the membrane. In-plane deformations in the x and y directions at the mid surface are zero.

00 xzyz

0xy

0z

00,,00,, yxvyxu

z=0

Free surfaceToo thin to have a reasonable Variation in z

5



xz

xz

zx

zx

x

zy

xz

xz

zx

zx

x

zy

The assumption regarding:

can be interpretted graphically in the following two figures:

One can easily see in the second figure that normals do not remain normal in the presence of xz strain.

0 zxxz

by a rigid body rotation

6



zy

yz

yz

zy

z y

x

The same graphical interpretation can be made regarding

0 zyyz

7



z

x yzzyxu ,,

xw

y

For Kirchoff plates displacements in the x-direction are assumed to vary linearly from the mid surface:

As a result strains in the x direction are characterized by the partial differenctial equation to the right:

2

2

xwz

xz

xu y

x

8



xzzyxv ,,

Simarlarly for displacements in the y direction:

which leads to:

Finally, the in plane shear strain is given by the expression

2

2

ywz

yz

yv x

y

yxwz

yxwz

xywz

xz

yz

xv

yu

xy

xy

2

22

2

9



yxw

yw

xw

xy

y

x

2

2

2

2

2

2

The curvature of the plates are defined as

Thus the strains can be expressed as

xyxy

yy

xx

z

zz

10



For plates the assumption of plane stress is made in the z direction. Thus the constitutive relationship is

where

xy

y

x

xy

x

x

D

2100

01

01

1 2

ED

11



• Note that the stresses vary linearly from the middle surface, just like bending stresses in beams.

• Also note that the shear stresses (xy) produced by bending also vary linearly from the middle surface.

• The shear stresses yz and zx are present and required for equilibrium, although the corresponding strains are assumed negligible. Parabolic variations of the stresses are assumed as in beam theory.

• The bending stresses can be simplified to resultant moments (Mx, My, Mxy). These moments are resultants of the linear stress variations through the thickness

12



x

zy

MxydyMxdy

Qxdy

QydxMyxdx

MydxThe moments Mx, My, Mxy (a torsional moment) as well as the shear forces Qxand Qy shown to the right are per unitl length quantities. 13



xy

y

x

xy

y

x

z

yxw

yw

xw

z

2

2

2

2

2

2

xy

y

x

xy

y

x

xy

y

x EzE

10001

01

1100

01

01

1 22

next page

2

2

t

txy

y

x

xy

y

x

zdz

dy

dxdy

dyM

dxMdyM

Here

where

14



xy

y

x

t

txy

y

x

t

txy

x

x

xy

y

x

Et

dzEz

zdz

M

MM

2100

01

01

112

2100

01

01

1

2

3

2

2

2

2

2

2

Thus

15



In a similar fashion

If we sum forces in the z-direction and sum moments about the x and y axes (Q and M are allowed to vary across the differential element, hence the use of gradients in the figure below and on the next page) assuming a pressure q is applied to the plate

2

2

t

t x

x

y

x dzQQ

qy

Qx

Q

qdxdydydxy

Qdxdy

xQ

qdAdxdyy

QQ

dydxx

QQdyQdxQ

F

yx

yx

yy

xxxy

z

0

16



In a similar fashion summing moments about the x-axis leads to

Similarly, summing moments about the y-axis yields (show for homework)

yyyx

yxyx

yyyxy

yy

xxx

Qy

Mx

M

dydxx

MM

dxdyy

MMdyMdxM

dyqdxdydydxdyy

QdxQ

dydxdyx

QdyQdydyQ

2

220

xyxx Q

yM

xM

0 17



Thus the equations of equilibrium at a point for a plate are

These equilibrium equations along with the moment-curvature relationships leads to the following non-homogenous fourth order, mixed partial differential equation in w that must be solved for plates:

If w is known, the strains are known. If the strains are known the stresses are known.

xyxy

yyxx

yx

Qy

Mx

M

Qy

Mx

M

qy

Qx

Q

0

0

0

qyw

yxw

xwEt

4

4

22

4

4

4

2

3

2112

18



Plate Element Formulation based on Kirchoff TheoryIn a paper published in 1984 Hrabok and Hrudley cited 88 types of plate element formulations. In this section one is presented and issues associated with the formulation are discussed.

Consider the basic rectangular plate element with four nodes each with three degrees of freedom, i.e., one out of plane displacement (w) and two in plane rotations (x and y). This element is depicted below. Shown in the figure are possible nodal forces (fwi) and nodal moments (fxi and fyi):

Keep in mind that unknown deformations (displacements and rotations) are teamed up with applied actions (forces and moments). 19



The nodal deformation vector at any node i is identified as

where

The negative sign on y is due to the fact that a negative displacement (w) is required to produce a positive rotation about the y-axis (recall the right hand rule). The total element deformation vector is

yi

xi

i

i

wd

xw

yw

yx

Tyxyxyxyx wwwwd 444333222111

20



Because there are 12 degrees of freedom for the rectangular element a 12 term polynomial in x and y is selected. A complete cubic polynomial has 10 terms.

We need two more terms from the quartic polynomial line of Pascal’s polynomial triangle. To understand how to pick these terms we need to discuss “Continuity Conditions.” Two types on continuity are considered. They are classified in the following manner:

C0 Continuity - The element is C0 compatible if w along any side of an element is completely specified by the degrees of freedom on along that side. Thus no gaps appear along the edge where two elements meet.

C1 Continuity - The element is C1 compatible if it is C0 compatible, and the slope the side is completely specified by the degrees of freedom on along that side.

termsothertwoyaxyayxaxa

yaxyaxayaxaayxw

3

102

92

83

7

265

24321,

21



With C1 continuity the slopes from two elements will match along an edge where the elements meet. This happens in the following depiction of elements viewed on edge:

Where in the follow figure slopes do not match along the edges of elements where they meet.

With Kirchoff plate theory, the slopes must match, giving a the smoother description of the deformed surface in the first figure.

22



If we add x3y and xy3 terms as the other terms in the polynomial for w then we get to the required 12 terms:

The element satisfies the transverse deflection continuity. Along an edge between two nodes, say the edge between nodes i and j in the figure below

the deflection w will vary in a cubic manner and y, a derivative of w, will vary quadratically.

3

123

113

102

92

83

7

265

24321,

xyayxayaxyayxaxa

yaxyaxayaxaayxw

23



That is for

relative to the local element coordinate axis, then

The constants a1, a2, a4 and a7 can be uniquely determined along this edge of the element by four degrees of freedom along that edge: wi, wj, θyi and θyj. Consequently C0

continuity will hold and apparently C1 is satisfied. But

37

24210, xaxaxaayxw

0y

2742

0

320, xaxaaxwyx

yy

311

2853

0

0,

xaxaxaa

ywyx

yx

24



is cubic in x, which requires 4 more boundary conditions to find a3, a5, a8 and a12. But only two more deformation conditions exist along this edge of the element, i.e., xi and xj. This slope is not uniquely defined and a slope discontinuity can occur along this edge. Hence this element does not satisfy C1 continuity conditions.

This does not mean that the constants a1 through a12 can not be determined. To see this consider that

12

3

2

1

3222

2322

33322322

302302010332020100

1

a

aaa

yyxyxyxyxxyxyxyxyx

xyyxyxyyxxyxyxyx

wwyww

y

x

25



With

or

12

11

10

9

8

7

6

5

4

3

2

1

344

24

2442

2444

244

34

2444

2444

3444

34

34

2444

24

34

2444

2444

333

23

2333

2331

233

33

2333

2333

3333

33

33

2333

23

33

2333

2333

322

22

2222

2222

222

32

2222

2222

3222

32

32

2222

22

32

2222

2222

311

21

2111

2111

211

31

2111

2111

3111

31

31

2111

21

31

2111

2111

4

4

4

3

3

3

2

2

2

1

1

1

302302010

332020100

1

302302010

332020100

1

302302010

332020100

1

302302010

332020100

1

aaaaaaaaaaaa

yyxyyxxyx

yxxyyxxyx

yxyxyyxyxxyyxxyx

yyxyyxxyx

yxxyyxxyx

yxyxyyxyxxyyxxyx

yyxyyxxyx

yxxyyxxyx

yxyxyyxyxxyyxxyx

yyxyyxxyx

yxxyyxxyx

yxyxyyxyxxyyxxyx

w

w

w

w

y

x

y

x

y

x

y

x

aCd 26



Finding the inverse of this relationship can be trivially stated as

With

where

dCa 1

aPw

y

x

3222

2322

33322322

302302010332020100

1

yyxyxyxyxxyxyxyxyx

xyyxyxyyxxyxyxyxP

27



Then

or

where

is the matrix of shape funtions.

dCPw

y

x1

dNw

y

x

1 CPN

28



212

211985

121096

11874

2

2

2

2

2

66442

66226262

2yaxayaxaa

xyayaxaaxyayaxaa

z

yxw

yw

xw

z

z

xy

y

x

xy

y

x

Returning the attention to the strains in a plate element

We can write

dCB

aB1

1

1

29



where

and

The element stiffness matrix is

221

660440020000606200200000060026002000

yxyxxyyx

xyyxB

11

CBB

dydxBDBtkc b

TT 2

0

2

0

30



There are many different ways of interpolating deformations over a plate, leading to many different types of plate elements having been proposed for Kirchoff’s plate theory. The same issue comes up for Mindlin-Reisner plate theory.

Some proposed elements for Kirchoff’s theory have encountered problems with locking -a refusal of the element to deform in some circumstances, as the set of equations becomes over-constrained, due to inconsistent assumptions being made. This is referred to as shear locking.

Other elements have had the opposite problem of lacking stiffness to resist certain “zero-energy” modes of deformation. These spurious modes of deformation then dominate the solution. Another name for this is “hourglass” deformation, as the unwanted in-plane deformation of a quadratic element resembles this shape.

31

section 13 plate elements.ppt

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