15. chapter 15 - plates and shels _a4

______________________Basics of the Finite Element Method Applied in Civil Engineering

137

CHAPTER 15 FINITE ELEMENTS FOR PLATES AND SHELLS Plates and shells are particular forms of three-dimensional solids, very often encountered in civil engineering structures (slabs/floors, diaphragm walls, reservoirs, roofs, etc). Plates and shells have planar or curved shape respectively. Their main characteristic is the reduced thickness t relative to the structural area. In finite element modelling, two approaches are possible:

- using C1 class elements, based on thin plate theory; in this case they define the middle plane (or middle surface) of the plate and fulfil some simplifying assumptions;

- considering the plates and shells as special cases of 3D solids, by using quadratic C0 class elements.

The basic assumptions for thin plates and shells are the following:

- the stresses in the normal direction to the middle plane (σz) are small enough to be neglected;

- the normal n to the middle plane remain straight and normal during and after deformation;

- the strains and the stresses have a linear distribution over the thickness;

- the deformations are small according to the geometrical dimensions. In thin plates theory, the in-plane displacements are usually neglected, while the deformed shape is completely defined by the normal displacement w(x, y). For plate elements subjected to pure bending, the in-plane forces are also disregarded.

Chapter 15 Finite Elements for Plates and Shells_________________________________

138

15.1 BASIC EQUATIONS In the differential approach, the plates’ equation has the general form:

022

2

2

2

2

=+∂∂

∂−

∂∂

+∂∂ q

yxm

ym

xm xyyx (15.1)

where mx and my are the bending moments per unit length, mxy the twisting moment per unit length and q is the distributed load per unit area, applied normal on the middle plane of the plate (see figure 15.1). In the general formulation, the strains are divided into in-plane strains due to bending (εx, εy, γxy) and strains due to transverse shear (γxz, γyz).

xθz

xu x

x ∂∂

−=∂∂

=ε ; yθ

zyv y

y ∂∂

−=∂∂

=ε ;

∂∂

+∂∂

−=∂∂

+∂∂

=xθ

yθz

xv

yu yx

xyγ

(15.2)

xwθ

xw

zu

xxz ∂∂

+−=∂∂

+∂∂

=γ ; ywθ

yw

zv

yyz ∂∂

+−=∂∂

+∂∂

=γ (15.3)

or, in matrix form

θε ⋅⋅−=

∂∂

∂∂

∂∂

∂∂

−=

= Lzθθ

xy

y

xz

γεε

y

x

xy

y

x

0

0

(15.4)

ww

y

xθθ

γγ

y

x

yz

xz ∇+−=

∂∂∂∂

+

−=

= θγ (15.5)


139

Fig. 15.1 Thin plate region with applied load, bending and twisting moments

The stress field of the plate is expressed by the so-called generalized stress vector σg with its components: the unit bending moments on both directions and the unit twisting moment.

==

xy

y

x

g

mmm

mσ (15.6)

Written on components

∂∂

+∂∂

−== ∫−

2

2

2

2*

2/

2/ ywµ

xwEzdzσm

t

txx ;

∂∂

+∂∂

−== ∫−

2

2

2

2*

2/

2/ xwµ

ywEzdzσm

t

tyy ; with

)1(12 2

3*

µEtE−

= (15.7)

yx

wµEzdzτmt

txyxy ∂∂

∂−== ∫

−

2*

2/

2/

1 )-( ;

Similarly, the unit shear forces are:

∫−

=2/

2/

t

txzx dzβτs ∫

−

=2/

2/

t

tyzy dzβτs (15.8)

)u(x)v(y

)w(z

q

xym

ym

xmxymxym

ym

xym

xm


140

Introducing the concepts of generalized strain,

∂∂∂∂∂

−

∂∂

−

=

yxwyw

xw

g

2

2

2

2

2

ε (15.9)

the stress-strain relationship due to bending is defined as

θ⋅⋅== LEEεσ gg (15.10) with the elasticity matrix for an isotropic, plane stress behaviour of the plate

−=

2100

0101

*

µµ

µEE (15.11)

As the conventional strains and stresses have linear distribution along the plate’s thickness, they can be calculated using the following expressions:

ztmx

x 312

=σ ; ztmy

y 3

12=σ ; z

tmxy

xy 3

12=τ (15.12)

where z is measured from the plate’s mid-plane and t is the thickness of the plate. Similarly, the strain – stress relationship due to the unit shear force yields

)( wtβss

y

x ∇+−=

= θGs (15.13)


141

where the βGt product defines the shear stiffness of the plate. The equilibrium equations are

0=+∂∂

+∂∂

xxyx s

ym

xm

0=+∂∂

+∂∂

yxyy s

xm

ym

(15.14)

0=+∂∂

+∂∂ q

ys

xs yx

or, in matrix form:

0smL =+=

+

∂∂

∂∂

∂∂

∂∂

T

y

x

xy

y

x

ss

mmm

xy

yx

0

0 (15.15)

0qsq =+∇=+

∂∂

∂∂ T

y

x

ss

yx (15.16)

with

∂∂

∂∂

∂∂

∂∂

=

xy

y

x0

0

L and

∂∂∂∂

=∇

y

x as differential operators.

In the peculiar case of thin plates (also called Kirchoff plates), the plane cross - sections normal to the middle plane are assumed to remain plane and normal to the middle plane during deformation. Thus, the shear stresses are suppressed (s = 0) and

0θ =∇+− w or w∇=θ (15.17)


142

For an isotropic plate of constant thickness, the equilibrium expression reduces to the bi-harmonic equation:

0)1(122 3

2

4

4

22

4

4

4

=−

+∂∂

+∂∂

∂+

∂∂

Etq

yw

yxw

xw µ (15.18)

15.2 FINITE ELEMENT APPROACH FOR PLATE BENDING The general variational principle can be written as the minimization of total potential energy associated to bending:

∫∫∫ −+=AA

TgA

Tg wqdAdAdAE(min) sγσε

21

21 (15.19)

where w is the lateral deflection of the plate and q the distributed load acting normal to the middle plane. The functional E is valid for general, thick plates. In the case of thin plates (Kirchoff plates), no transverse shear occurs, i.e. γ = 0. The variational principle yields:

∫∫ −=AgA

Tg wqdAdAE(min) σε

21 (15.20)

When defining the strain and stress fields, the functional expression contains second order derivatives of the unknown w(x, y), thus, a C1 class of continuity should be satisfied. Both displacement function and its derivatives should be continuous across elements boundaries, along the normal and the tangential directions. The elements’ DOF are the nodal

displacements w(x, y) and their first degree derivatives xwθ x ∂∂

= and

ywθ y ∂∂

= .

For thin plates ( w∇=θ ) the variables are expressed in terms of normal displacement w


143

wg ∇== LLθε ; wgg ∇== ELEεσ (15.21) and the functional will be also expressed in terms of w only:

( ) ( ) ∫∫ −∇∇=AA

T wqdAdAwwE LEL21(min) (15.22)

Expressing the nodal values of the unknown as

[ ]Tiyxi w θθ=a (15.23)

the approximation yields:

w = N⋅a (15.24)

It follows that aBNaLLε ⋅=∇⋅=∇⋅= wg and ∑=n

eEE1

where

( ) qdAdAEA

TT

A

TTe ∫∫ −= NaaEBBa

21 (15.25)

or,

rakaa TTeE −=

21 (15.26)

From this point forward, the standard procedure can be followed. 15.3 FINITE ELEMENTS FOR THIN PLATES BASED ON IRREDUCIBLE THIN PLATE APPROXIMATION As it was shown, the procedure is almost standard, once the shape functions N are chosen. In fact, choosing the shape functions stands for the main difficulty of the problem. 15.3.1 Continuity requirement for shape functions Not only a C1 class of continuity is required but also a physical continuity of the lateral displacement and slope across elements interface. This can be


144

achieved only if w and nw ∂∂ are uniquely defined by the nodal parameters along such an interface. It can be demonstrated* that it is impossible to specify simple polynomial expressions as shape functions ensuring full compatibility, when only w and its slopes are used as nodal parameters. The attempts to associate the second derivatives 22 xw ∂∂ , 22 yw ∂∂ and

yxw ∂∂∂ 2 as nodal parameters apparently solve the compatibility, but violates other physical requirements, creating an “excessive” continuity. If, for example, two adjacent elements (1) and (2) have different material properties at the interface (E1 ≠ E2) or different thicknesses (t1 ≠ t2), the

condition 2

2

2

12

2

xw

xw

∂∂

=∂∂ means an excessive continuity because

21 xx mm ≠ for 2

23

)1(12 xw

µEtmx ∂

∂−

= .

Thus, the physical difference at the interface between elements curvatures is not longer truthful. 15.3.2 Conforming and non – conforming shape functions The difficulty of finding compatible displacement functions led to the option of disregarding the complete slope continuity. Thus, two types of elements are developed:

- elements that enforce slope continuity only at nodes (3 DOF/node, w, xw∂∂ ,

yw∂∂ ), hoping to achieve, at limit, a complete slope continuity. They are

called non-conforming elements. The convergence of such elements is not certain but can be verified by numerical tests (as the patch test). Some of them are quite successful but, due to their uncertain behaviour under various conditions, they are usually not included in the general purpose modelling codes;

- elements which use the same nodal variables (w, xw∂∂ ,

yw∂∂ ), but the slope

compatibility is achieved by an additional shape function corresponding to a

* Zienkiewicz, O. C.- The Finite Elements Method, Mc Graw – Hill, 1979


145

new variable which is the normal slope nw∂∂ at the mid-side point. Such plate

elements are called conforming elements. However, in order to avoid the assembly difficulties, the mid-side node degrees of freedom can be constrained. After suitable transformations, the result is a compatible element with exactly the same number of DOF per element as the non-conforming one. In the following, the Clough-Tocher triangle and its corresponding Clough-Felippa compatible quadrilateral element will be presented. Being very popular in structural analysis, they are included in well known computer codes. Others finite elements are also available but the basic principles are generally the same. 15.3.3 Liner curvature compatible triangle LCCT-12 Clough – Tocher In order to achieve complete compatibility between plate elements, it was demonstrated* that the approximation of the displacement function should be expressed as a complete cubic polynomial:

310

29

28

37

265

24321 yaxyayxaxayaxyaxayaxaa)y,x(w +++++++++=

(15.27)

For triangular elements with 3 DOF/node, only 9 equations are available, while giving up a polynomial term or using symmetry as 98 aa = leads to non-conforming shape functions (for which the inverse of the CM matrix is sometimes impossible to calculate). For this reason, the Clough - Tocher element is made of three sub-triangles, connected in an interior node P, placed in the centre of the element. For each sub-triangle, a cubic approximation such as (15.27) is used, the total number of polynomial

coefficients being 30. The element has 12 DOF, w,xw∂∂ ,

yw∂∂ for nodes 1, 2,

3 and only nw∂∂ for nodes 4, 5 and 6. In order to determine the polynomial

coefficients, the following boundary conditions are equated, leading to an algebraic equations system 30 × 30:

* Zienkiewicz, O. C.- The Finite Elements Method, Mc Graw – Hill, 1979


146

- the nodal values at the corner nodes of the element, taking into account the connection of two sub-triangles at each corner node;

- the normal rotations nw∂∂ at mid-side and interior nodes 4, 5, 6 and 7,

8, 9 respectively; - the continuity of displacements and rotations of the interior node P.

Fig. 15.2 The Clough – Tocher triangular finite element

Although it requires many algebraic calculations, the element is very effective in practice, accomplishing a linear variation of the curvature vector [ 22 xw ∂∂ 22 yw ∂∂ yxw ∂∂∂ 2 ]T on each sub-triangle. The main disadvantage is the presence of the mid-side nodes 4, 5 and 6. They can be eliminated by prescribing a linear variation of the curvature along each side, leading to alternative elements LCCT-11, LCCT-10 or LCCT-9. A quadrilateral plate element can be obtained by assembling four LCCT-11 elements (the so called Clough-Fellipa element). This element has 19 DOF, from which 7 are associated to interior nodes (see figure 15.3). After eliminating the interior DOF by static condensation, the element finally has 12 DOF (w, θx, θy for nodes 1, 2, 3 and 4).

1

2

3

1

2

3

4

5

6 P

I

II

III

7

8

9

( )yx ,,w θθ

( )nw∂

∂

x

y


147

Fig. 15.3 The Clough – Fellipa quadrilateral finite element 15.4 FINITE ELEMENTS FOR SHELLS The behaviour of thin structures with a continuous curvature can be reasonably approximated by meshing them into small, flat polygonal surfaces. By reducing the elements dimensions and increasing the elements number respectively, the solution of such a discrete model converges to the curved shell’s solution. A shell element, part of a thin curved structure, withstands the applied load by both bending and membrane effects. The membrane effect is the one inducing in-plane stresses, due to the u and v displacements (expressed in the local coordinate system). Consequently, the total potential energy is written as the sum of these effects:

−+

−= ∫ ∫∫ ∫ V Vg

TgV V

TTp wqdVdVfdVddVE σεσε

21

21 (15.28)

The first group of terms inside the brakes defines the deformation energy due to in-plane strains and stresses, while the second one the deformation energy due to generalized strains and stresses (curvature and unit bending moments). Initial strains and stresses are neglected.

1

2

3

4

5

6

I

II

IV 7

8

9

x

yIII

( )yx ,,w θθ

( )nw∂

∂


148

Fig. 15.4 The Shell - type quadrilateral finite element

For the finite element shown in figure 15.4 the attached local coordinate system is ( z,y,x ), with the z coordinate normal to the elements mid-surface. The DOF are u and v for the membrane effect and w, θx, θy for the bending effect. The finite element solution leads to the following expression for the total potential energy:

−+

−= b

Tbbb

Tbm

Tmmm

Tme,pE rδδkδ

21rδδkδ

21 (15.29)

where [ ]411 v...vuT

m =δ is the displacement vector for the plane stress state of the element, mk the stiffness matrix in the same hypothesis and

mr the nodal load vector with in-plane components. The bending DOF are [ ]42111 ,y,y,x

Tb ...ww θθθ=δ , the bending stiffness matrix is bk and

the corresponding applied load vector is br (with its components, the normal loads q and the prescribed unit moments xm and ym ). The energy expression emphasizes that the plane stress displacements are not affecting those due to the bending effect and vice-versa. Because the rotation zθ does affect neither the deformation nor the energy, in order to have a unitary definition, a fictitious moment zm is attached. All matrices and vectors are augmented with zero terms reaching the total number of DOF. Subsequently, they are summed to the global expression:

mbTmbmbmb

Tmbe,pE rδδkδ −=

21 (15.30)

xFyF

)u(x)v(y)w(z

xMyM

xθyθxy

z


149

Because the stiffness matrix and the characteristic vectors are expresses in local coordinates, a transformation operation is needed in order to add all elemental contributions to the total potential energy of the structure. From this point forward the solution follows the standard procedure. 15.5 PLATES AND SHELL ELEMENTS AS THREE-DIMENSIONAL SOLIDS Plates and shells are peculiar cases of 3D solids, thus, they can be meshed using 3D finite elements. Because of the curved geometrical shape of shells, the most suitable option is the quadratic isoparametric finite element, a C0 class hexahedron shown in figure 15.5.

Fig. 15.5 Quadratic hexahedral finite element for shells

In order to define the element’s matrices, the main difficulty encountered is due to the relative reduced thickness t versus the other mid-surface dimensions of the element. The matrix terms corresponding to the thickness direction ( z ) are with several orders of magnitude larger then the other terms. Thus, because the global equation system RKδ = is bad conditioned, the result may be erroneous. Another possible problem is due to the large number of DOF (20×3 = 60 DOF), which is in direct relationship with the amount of required virtual memory and computing time. Moreover, the third mid-node placed on the thickness direction violates the basic hypothesis regarding the straight shape of the normal to the element’s mid-surface. By contrary, the hexahedral

xy

zt

a

b


150

element has the advantage of modelling the shear deformation due to transverse tangential stresses, which was not possible with the elements presented before. Several attempts where made to reduce the element’s DOF, by reducing the nodes number using static condensation. Such a finite element is represented in figure 15.6.a. The mid-edge nodes along the thickness direction are eliminated using kinematical restrictions. In order to improve the element’s behaviour in bending, a supplementary interior node is attached. The shape function matrix N, with only 48 available DOF, is completed with the relationships regarding the interior node, such as

( ) ( )βα 2217 11 ts)t,s(N −+−= (15.31)

while for the eliminated mid-edge nodes a straight line positioning condition is applied:

( ) ( ) ( )[ ]tbm w,v,uw,v,uw,v,u +=21 (15.32)

where u, v, w are the displacement DOF and the indexes m, b and t stands for middle, bottom and top.

Fig. 15.6 Simplified procedures applied for 3D shell finite elements

Another option is to express the 16×3 displacement DOF corresponding to the elements’ top and bottom faces as DOF - displacements and rotations –

s st t

r r

1

2

3

4

5

6 7

8 9

10

11

12 13

14 15

16 17

1

2

3

4

5

6 7

8

9

2’’

2’ 4’

4’’

a. b.


151

of 9 mid-surface nodes, as it is shown in figure 15.6.b. The mid-surface displacement is defined as the mean value of the equivalent top and bottom nodes, while the rotation is expressed as the difference between the top and bottom displacements divided with the element thickness t. Another DOF is eliminated by disregarding the normal stress rσ . The stiffness matrix is finally defined by 40 external DOF which means a 33% reduction relative to the equivalent linear quadratic hexahedron.

15. chapter 15 - plates and shels _a4

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