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A STUDY OF THE APPLICATION OF MINDLIN
PLATE ELEMENTS TO THIN PLATES
by
Piyoros Jirawattana
B.Eng., Chiang Mai University, 1994
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
1997
r·-L.~.Li
This thesis for the Master of Science
degree by
Piyoros Jirawattana
has been approved
Date
Jirawattana, Piyoros (M.S., Mechanical Engineering)
A Study of the Application ofMindlin Plate Elements to Thin Plates
Thesis directed by Assistant Professor Samuel W. J. Welch
ABSTRACT
This thesis is a study of Mindlin plate theory in finite element analysis with particular emphasis on locking and instability behaviors of different element formulations. A computer model is used to simulate the bending action of thin plates and to study locking and instabilities associated with the different element formulations.
We start with the bilinear Mindlin plate element. Using techniques of numerical integration we demonstrate that "shear locking" can be eliminated and a very competitive convergence results. This is done by using a technique known as selectivereduced intergation. However, the selective-reduced integration method can lead to an unstable mode in the finite element solution.
Finally, we consider the development of a Mindlin plate element that does not lock and does not contain unstable modes. This is done by approximating the transverse shear strain using shape functions called mixed interpolation functions.
The detection of locking and instability will be presented by imposing spatial Fourier modes and comparing the strain energy from finite elements to the strain energy of the continuous Mindlin plate theory. This idea can also be used in the analysis of convergence of the solution of finite element analysis.
This abstract accurately represents the content of the candidate's _thesi,¥' I recommend its publication.
Signed
,. / Samuel W. !· W lch
iii
ACKNOWLEDGMENTS
I would like to thank Samuel W. J. Welch for the thesis topic and his guidance. I also wish to thank John A. Trapp for the fundamental plate theory from advanced strength class.
iv
CONTENTS
Acknowledgments ............................ .... .... .... ........ .... ......................................... .... iv
CHAPTER
1. INTRODUCTION AND PLATE THEORETICAL PRELIMINARIES ................ 1 Scope ................................................................................................................. 1 Kirchhoff and Mindlin Plate Bending Theories .................................................. 2
Introduction .................................................................................. ................ 2 Stress-Strain Relationship for Flat Plate ............................................................ 2
Kirchhoff Theory .......................................................................................... 3 Moment-Curvature Relation ................... .......................... ....... .... .... ......... 6
Mindlin Theory ............................................................................................. 7 Moment-Curvature Relation .................................................................... 1 0
2. FINITE ELEMENTS FOR PLATES .................................................................. 11 KirchhoffPlate Elements ................................................................................. 11 Mindlin Plate Elements .................................................................................... 13
Stiffness Matrix .......................................................................................... 14 Discussion of Kirchhoff and Mindlin Plate Elements ....................................... 16
3. INSTABILITY AND LOCKING ...................................................................... 18 Introduction ......................................................................................... ........... 18 Locking ............................................................................. ............................. 19
Locking in Mindlin Beam ........................................................................... 19 Numerical Examples Using Mindlin Plate Elements ...................................... 23
Instability ........................................................................................................ 27 Zero Strain Energy in Mindlin Beam .......................................... ................ 27 Eigenvalue Analysis of the Zero Strain Energy Modes ................................ 30 Analysis oflnstability and Locking ofBilinear Mindlin Plate Elements Using Spatial Fourier Modes ...................................................................... 31
Typical Instability Mode in Mindlin Plate Element ....................................... 38
4. THE DEVELOPMENT OF A MINDLIN PLATE ELEMENT ......................... .41 Approach ....................................................................................................... 41
v
Numerical Examples .................................................................................... 46 The Accuracy ofthe Mixed Interpolation Element ...................................... .48
5. CONCLUSION ., ........................................................................................... 50
APPENDIX
A. Stress Resultants in a Flat Plate ................................................................... 51 Force and Moment Balance ofFlat Plates .................................................... 55
Force Balance for Plates ........................................................................... 55 Moment Balance for Plates ....................................................................... 56
B. Derivation ofFour-Node Quadrilateral Mindlin Plate Element ....................... 58 Numerical Integration .................................................................................. 65
Two Dimensional Integrals ...................................................................... 65 Stiffness Integration ................................................................................ 66
REFERENCES .......................................................................................... .. ..... 68
vi
CHAPTER 1
INTRODUCTION AND PLATE THEORETICAL PRELIMINARIES
Scope
Plate bending has been studied in the field of classical strength of materials for a long time. The problem of plate bending can be considered as an extension of either elementary beam theory or Mindlin beam theory depending on which theory is appropriate. Both plates and beams support transverse loads by bending action. Two theories of classical plate bending will be discussed, Kirchhoff theory and Mindlin theory. The difference between these plate bending theories can be classified by the magnitude of the thickness compared to the magnitude of the other dimensions and also the magnitude of the transverse displacement compared to the thickness. The plate in this context will be considered as a relatively thin plate with small deformation.
The classical Kirchhoff plate bending theory makes an assumption that the transverse shear deformation of thin plates in bending can be neglected. Information about the resulting deformation can be assumed to be contained in the function w(x~,x2) and in it's derivatives. Therefore, finite elements based on this theory must use C1-continuous elements. Even though this classical theory is applicable to thin plates, it's use in finite elements is problematic as it requires higher-order nodal interpolation function to enforce inter-element compatibility. These higher order interpolations can be quite complicated and impractical for general use.
In Mindlin plate bending theory, the transverse shear deformation is taken into account so that this theory is applicable to thick plate bending. The deformation of Mindlin plate theory consists of one transverse deformation and two small angle rotations. The resulting finite element implementation requires only CJ -continuous elements so that lower order interpolation functions may be used. In addition, because the deformation in Mindlin plate theory is more physical, it is easier to apply boundary conditions. However, the application of Mindlin plate elements to thin plates may not be as accurate as Kirchhoff plate elements because the transverse shear strains from Mindlin theory, if approximated inaccurately, may make thin plates
too stiff. We will see that by the use of special techniques, Mindlin plate elements can be made accurate for thin plate problems.
Thus, the context of this study is the application of Mindlin plate elements to the bending of thin plates. We will begin with general background theory of Kirchhoff plate bending and Mindlin plate bending in chapter 1. Next, the application of both theories in finite elements will be discussed in chapter 2. Chapter 3 introduces the concept of "shear locking" and how methods to prevent this problem result in the side effect known as "instability". Finally the development of a Mindlin plate element for thin plates that does not lock and does not exhibit instability will be outlined in chapter 4. The numerical results presented in this study were obtained using software written by the author. The software is not included as part of this thesis.
Kirchhoff and Mindlin Plate Bending Theories
This section introduces, discusses and compares the Kirchhoff and Mindlin plate bending theories. A more comprehensive discussion of these theories is contained in reference [ 15] and in the references therein.
Introduction
A flat plate supports transverse loads by a bending action similar to a straight beam but the deformation modes are two dimensional and thus are more complicated. Plate theories assume that the plate thickness is relatively small compared to the plate dimensions in the plane of the middle surface. Thus the bending behavior of a plates in both Kirchhoff and Mindlin theory depends on the plate thickness. The former theory makes the assumption that straight fibers normal to the plane of the middle surface remain normal to the plane of the middle surface after bending. The latter theory does not make this assumption. Before discussing the details of these theories, we will start by discussing the general stress strain relationships for flat plates.
Stress-Strain Relationship for Flat Plate
The stress-strain relationship may be obtained from Hooke's law for isotropic elasticity.
2
(I. I)
In both Kirchhoff and Mindlin theories CT33 is considered negligible. From equation (I. I) with CT33 = 0, we obtain
(I.2)
substituting this into Hooke's law we obtain
(I.3)
equation (1.3) is the general stress-strain relation for a flat plate and from equation (I. I) we get the equation for transverse shear stress
CTaJ = 2GEa3 (1.4)
where a and Prange from I to 2 , 8ap= I if a= P and 8ap = 0 if a :t: p
Kirchhoff Theory
iW u1 =x3 -
-------------~----_ ......... ... ...... --------
Figure 1.1 Kirchhoff plate after loading
3
In Kirchhoff theory, it is assumed that the straight-line normals to the undeformed middle plane of the plate remains straight and inextensional under the deformation of the plate. In addition, it is assumed that the normals in the deformed plate remain normal to the deformed middle plane which is a reasonable assumption for thin plates. Under this assumption, it follows that EaJ = 0 which implies that aaJ = 0, hence the shear forces Q1 and Q2 are implied to be zero. Note that aaJ is neglected in the stress strain laws but Qa is kept in the equations of motion. Despite this contradiction, the theory is more accurate than the membrane theory of plates [6].
If we expand transverse displacement u3(x1,x:z,x3) about the mid plane using power senes m x3 we get [ 14]
X ifthe plate is thin - 3 << 1 and we can approximate
I
from the assumption that Ea3 = 0 we get
(1.5)
and if the plate is thin we can introduce equation (1.5) into the equation above, then
integrating with respect to x3 we get
(1.6)
Ua(x1,x2 ) is constant of the XJ integration which has no effect on the moment Map
and thus may be neglected. The resulting kinematics assumptions for Kirchhoff theory (neglecting 0::~3, 0'13 and a23 in the stress-strain law) are
4
(1.7)
(1.8)
(1.9)
1 (iii iii J from & all = - _a + _fJ we get 2 tJxp tJxa
(1.10)
(1.11)
(1.12)
Equations ( 1. 7) and ( 1. 8) are the kinematics assumptions of Kirchhoff theory mentioned earlier. Considering equations (1.7), (1.8) and (1.9), we can see that the deformation in x1 and x2 direction are related to w by first order differential equations. This leads to the second-order differential equations for strain in equations (1.10), (1.11) and (1.12).
To get the expression for bending moments in term of the transverse displacement, we substitute equations (1.10), (1.11) and (1.12) into equation (1.3), and then use the definition of bending moments (see the details of this definition in appendix A)
to obtain
h
2
M afJ = J x3u afJdxJ h --2
5
(1.13)
(1.14)
(1.15)
where D is the flexural rigidity (define in the next section) which is approximately
equivalent to E times the moment of inertia of a unit width plate.
Moment-Curvature Relation. In Kirchhoff theory we neglect u33 and also transverse shear strain, so that we can consider the in plane stress-strain relation. By assumption the material is isotropic. Let x1, x2 be the principle axis of material then we can write
v 0
v 1 0 0 0 (1- v)
2
(1.16)
We obtain the moment-curvature relation by substitution of equations (1.1 0), ( 1.11) and ( 1. 12) into equation ( 1.16) and the result into the definition of bending moments, yields [ 12]
vD
D
0
6
(1.17)
Eh 3
where D = 2
, E is the elastic modulus and v is Poisson's ratio. 12(1 - v )
Mindlin Theory
In Kirchhoff theory normals to the midsurface remain normal during deformation and hence transverse shear deformation is neglected. Mindlin theory makes an attempt to keep the transverse shear deformation which by assuming that fibers originally normal to the midsurface can rotate independently ofthe normal direction.
Ut=-x (} 3 ....
-------------------------------------,'
Figure 1.2 Mindlin plate after loading
iW We define the transverse shear strain, y =-- (} and note that in Kirchhoff a
theory this transverse shear strain is zero. In Mindlin plate theory we assume that the deformation in the x1 and x2 directions is a function of small angle rotations so that we can write,
7
(1.18)
( 1.19)
(1.20)
Equations (1.18) and (1.19) have included some rotation of fibers originally normal to the midsurface to orientations not normal to the midsurface direction. To obtain the expression for strains in Mindlin theory, introduction of equations ( 1.18), ( 1.19)
and (1.20) into the definition of E;J =_!_(iii; + iUJJ yields 2 Ox} Ox;
_ ((}a.fJ+(}fJ.a) EaP- -x3
2
and E = _! ( CU 3 + CYI a ) = _!_ (~ _ (} ) a
3 2 Ox Ox 2 Ox a a 3 a
from equations (1.21) and (1.22) we can write the expression for strains as
where (}a •fJ = Of} a 0(}/l
8
(1.21)
(1.22)
(1.23)
Considering equation ( 1.22), C1a3 is independent of x3 direction, so it is constant over the plate thickness. In reality CTa3 has a parabolic profile over plate thickness. Instead of developing a very complicated theory to approximated CTa3 in parabolic manner with x3 (zero at h'2,-hl2. and maximum at x3 = 0). A k factor is added in the transverse shear force constitutive equation. We can evaluate the correct k shear correction factor by forming the strain energy for plate theory corresponding to a 3-dimensional energy with a parabolic shear stress.[14]
To get the expression of bending moment and the transverse shear force in Mindlin theory in terms of the displacement function, substitute strain equations (1.23) into the stress-strain relations (1.3) and (1.4). Then using the relationships
and
h
2
Map = I X3C1 apdx3 h
2
h
we obtain
M --h --8 B +B + 2G--+-3 [ vE { ) ( 1 ( 00 a Of) p JJ]
afJ - 1- v2 a/1 .... ..... rl 'rl 2 Oxp Oxa (1.24)
(1.25)
It should be noted as transverse shear strain has been included, the transverse shear stress can be written as
{1.26)
9
Moment-Curvature Relation. Let x1 and x2 be the principle axis of the material. The Moment-Curvature relation for Mindlin theory includes the shear stress-strain relations 0"23 = Gy23 and 0"31 = GyJJ. The Moment-Curvature relation is abbreviated as {M}=-[Du] { K}. Again, we assume that the material is isotropic, from equations (1.24) and (1.25) we can explicitly write
Mil 0 0 0 (} x. '.r•
M22 [DK] 0 0 0 (} %1 'Zl
Ml2 = 0 0 0 (} +B .1'1 'Zl %2 'Zt (1.27)
Q2 0 0 0 Gh 0 e.r2 - w,.r2
Ql 0 0 0 0 Gh e.r -w,.r I I
Where [DK] is the same as equation (1.17). Compare equation (1.27) with equation (1.17), as we mentioned ,Kirchhofftheory neglects transverse shear deformations so that in Kirchhofftheory the transverse shear forces Q1 = Q2 = 0.
10
CHAPTER2
FINITE ELEMENTS FOR PLATES
The preceding chapter is devoted to the analytical derivation of Kirchhoff and Mindlin plate bending theories. We might observe that numerical approximations of Mindlin theory should be easier because the governing equations are of lower order. We will discuss finite elements using these two theories.
Kirchhoff Plate Elements
As we have discussed in Chapter 1, Kirchhoff theory neglects transverse shear deformation and is applicable to thin plates. Strain energy is determined by in-plane strains &11, &22 and YJ 2. From equation (1.3), strains are determined by the lateral displacement field w = w(x1,x2). Nodal d.o.f. of Kirchhoff elements consist of lateral deflections w, and small rotations w,... and w,... of midsurface normals. We now
I 2
express the displacement field in terms of independent shape function interpolations as [12]
(2.1)
(2.2)
(2.3)
In order to formulate an element stiffness matrix [k]e, we now form the strain energy.
(2.4)
These strains of Kirchhoff theory in equations (1.10), (1.11) and (1.12), may be written in matrix notation as
II
4
' ~' • .x2
Figure 2.1 Plate 12 d.o.f. at node 3 of Kirchhoff plate element
We consider that the thickness constant over the plate, so that we have dV = hdA and finally dA = dxdy. Now using equation (1.16), equation (2.4) can be rewritten as
(2.5)
where {K"f = [w,.r1.r
1 w,.r
2.r
2 2w,.r,.r
2] is known as the curvature for Kirchhoff
elements and [DK] is known as the flexural rigidity
From equations (2.1), (2.2) and (2.3), we can write the nodal displacement matrix of a Kirchhoff element as
12
{d} = [w1 w,.r 1 w,.r 1 I 1 (2.6)
Equation (2.6) can be written via the nodal interpolation shape functions as
w = [N]{d} (2.7)
equation (2. 7) allows us to write the curvature using nodal shape functions as
{I(}= [B]{d}
Where [B] is the differentiation matrix that contains information about the curvature.
Substituting equation (2. 7) into equation (2. 5), we obtain the element stiffness matrix,[k]
(2.8)
where [k] = j[Bf[DK ][B]dA A
Mindlin Plate Elements
Mindlin plate theory includes transverse shear deformation, therefore it is a theory that is reasonable for thick plates.
Nodal d.o.f. of Mindlin plate elements consist of one lateral deflection w, and two
rotations, (} .r 1 and (} .r 1 of midsurface normals. If all interpolations use the same I 1
polynomial, then for an element of N nodes, we can write
(2.9)
where (2.1 0)
13
Consider equations (1.23) and (2.9), the field for w is coupled to the fields for 8z I
and 8z only via the transverse shear strains 'Y23 and ~ 1 [12]. 2
4 3
w 8
1 2 z, ~.-X-~----------------------~~~~~
Figure 2.2 12 d.o.f at nodes 2 ofMindlin plate element.
Stiffness Matrix
In a like manner as we did for the Kirchhoff plate element. We can write the formula for the strain energy U [ 12]
h
1 2 U =-J J {e}[E]{e}ch3d4
2 A h 2
14
(2.11)
where A is the area of the plate mid surface and
{ E} = [c X X EX X r X X r X X r X- r ] . Expressions for each strain are stated in I I l l I l l l rl
equation {1.23). Integration through the thickness yields
(2.12)
where [DM] and {K'} are stated in equation (1.27). We can write
{K}= [L]{u} (2.13)
0 0
0 a-1
0 0 0
a-2 where 0
0 0 [L]=
a-2 a-1 0
0 &2 0
1 0 a-1
Equations (2.9) and (2.13) yield
{K} = [B]{d}
where
15
0 N":r.. 0
0 0 Nl•.r l [B] = [L][N] = 0 Nl•.r Nl•.r (2.14) l I
-NI•.rl 0 Nl ...........
-Nl•.r Nl 0 ··········· I
And finally from equations (2.12) and (2.14)
where
Discussion of Kirchhoff and Mindlin Plate Elements
Sometimes Mindlin theory is called thick plate theory and Kirchhoff theory is called thin plate theory. Consider the d.o.f in the displacement matrix, {d}, of
Kirchhoff and Mindlin plate elements. In Kirchhoff theory, the d.o.f at node i are one lateral displacement w, and two rotations, w,.r,• and w,.r:t. In Mindlin theory, the
d.o.f at node i are one lateral displacement w,. and two rotations (}.r, and (}.r, We I l
can see that the displacement matrix of Mindlin has lower order partial differentials that make it more physical and easier to apply numerical approximations. In addition, the higher order partial differentials in Kirchhoff theory requires at least biquadratic shape function that lead to complicated numerical evaluation. Mindlin theory needs only bilinear shape functions.
Considering equations (1.7), (1.8) and (1.9), Kirchhoffplate elements must display interelement continuity of w, w •.r and w •.r (compatibility). Consider the connection
I l
between elements in the x2 direction, continuity of w forces the continuity of the
tangential slope, w,.r but not continuity ofthe boundary-normal slope w,.r . Now for l I
Mindlin theory, consider equations (1.18), (1.19) and (1.20). Compatibility requires the continuity of the fields, w , (} .r and (} .r at the interelement boundaries. Mindlin
I l
plate elements are easier to enforce compatibility because the lateral displacement fields are not related to the rotation fields by differentiation.
16
In the next chapter, we will show that by using selective-reduced integration, Mindlin plate elements can capture behavior of thin , or Kirchhoff plates. These are the reasons that Kirchhoff plate elements are not widely used.
17
CHAPTERJ
INSTABILITY AND LOCKING
Introduction
So far, we have discussed the derivation of Mindlin plate theory and it's implementation in finite element theory. In this chapter we compare results obtained after selective-reduced integration and full integration are used for the integration of the element shear stiffness matrix. This leads to the study of two important concepts in finite element analysis that can make practical results unrealizable, instability and locking. Numerous researchers have made attempts to prevent these problems.
p
Figure 3.1 Cantilever beam
18
Locking
Mindlin plate theory make plate bending theory more physical by including transverse shear deformations into account. However in finite element analysis approximation of these transverse shear strains can lead to inaccurate results. Physically the deformation of thin plates should be dominated by the bending deformation but if it is dominated by transverse shear deformation we call this "shear locking".
Locking in Mindlin Beam
To explain how this shear locking occurs we consider the cantilever beam of length L with load P in Figure 3. 1 and we use two different integration methods, full integration and selective-reduced integration.
We employ Mindlin plate theory with the nodal d.o.f w~, ~. w2, Oz. We separate the stiffness matrix [ k] into a component with bending stiffness [ k b ] and a
component with transverse shear stiffness [ k.] . We do so in order to integrate the
bending and the shear terms separately.
We obtain [Bb] that is associated with in-plane strain
B _ [0 -1 I L 0 1 I L] [ b]- 0 0 0 0 (3.1)
and [B.] that is associated with transverse shear strain
[ 0 0 0 0 ] [B] =
• -11L xiL-1 IlL -xiL (3.2)
case 1) Full integration
Introduction of equations (3 .1) and (3 .2) into the expression for the stiffness matrix and using exact integration for both the bending stiffness [kb] and the
transverse shear stiffness [k.] we obtain the stiffness matrices
19
I
L
[k.] =~A 2
-I
L 2
0 0
0 -I
0 0
0 -1 0
L -I
L 2 2
( ~2) L (~) 2 L
I L
2 2
(:) L ( ~2) 2
With k = ~ , we solve for the tip displacement 6
12(%r + 20
(6PL)
(h) 2 5GA
I2 - +5 L
for a thick beam, h >> L , the above solution becomes
6PL w ==--
2- 5GA
(3.3)
(3.4)
(3.5)
(3.6)
Equation (3.6) is a reasonable result for a thick beam with deformation caused by shear. Compare equation (3.6) with the deformation caused by shear from
Castigliano's theory, w 2 = 1.33 PL [IO]. Thus, the Mindlin beam provides a AG
reasonably accurate solution for thick beams.
20
Now for a thin beam, h << L, from equation (3.5) we obtain
24PL w =--
2- 5GA (3.7)
In the latter case the deformation should be dominated by bending (compare to the
analytical solution from elementary beam theory w2 = PL3
) but it is not. It is 3£/
dominated by shear. This is an example of shear locking. Now we consider what happens if selective-reduced integration is applied.
Case 2) Selective integration
In this case, we integrate bending stiffitess with two Gauss points and use one Gauss point for the shear stiffitess. This yields the same bending stiffitess as before but we get a different shear stiffitess.
1 L
-1 L
2 2 L (~) L (~)
[k.] =~A 2 2 (3.8)
L L -1 1
2 2 L (~) L (~) 2 2
Solving for the deflection at the free end, we get
(3.9)
The first term and the second are deformation from shear and bending, respectively. When the beam becomes thin, the second term contributes and we can neglected the first term. The solution from elementary beam theory (using Poisson's ratio of 0.3 )
21
2PL3
IS w = Selective-reduced integration with a Mindlin beam finite element 2 1.3Gbh3 .
produces a result of 97.5 % accuracy. It should be noted that for a thick beam solution, either full integration or selective-reduced integration can capture the behavior of shear deformation because the bending deformation can be neglected.
It is counter to our intuition that an exact integration produces an inaccurate result but an under-integration might introduce an acceptable answer. In order to explain, we consider the cantilever beam in Figure 3.1, obtained by simply reducing a Mindlin plate element into one dimension. We can write the equation for axial strain c 11 and transverse shear strain r 31 as
(3 .1 0)
(3 .11)
where the notation is the same as that of the Mindlin plate element.
We separate the element strain into the bending strain and the transverse shear strain. We can write
(3.12)
Our discussion will be concerned with the transverse shear approximation in equation (3.12).
(3.13)
If we consider equation (3.13) we see that the terms associated with the transverse deformation are constant. The terms associated with the rotational degree of freedom are linear in x. If we use selective-reduced integration we will evaluate (3 .13) at x =
L/2, we are in effect replacing (3. 13) with
(3.14)
22
In the literature it is common to see the above process referred to as eliminating the excess monomial. This excess monomial results from using inappropriate order polynomials for the displacement and rotation fields.
When the beam becomes thin, the transverse shear terms should become infinitesimal. The shear resulting from evaluating equation (3.13) at two Gauss points cannot, in general, be infinitesimal at both points and the resulting error becomes magnified in thin beams as the stiffuess matrix divides this error by the thickness squared. Thus, in the case of thin beams, if full integration is used with the shear stiffuess matrix, the deformation will be dominated by [ k sl not by [ k b ] . This is called
"Locking". In the literature the use of full integration on the shear stiffuess terms is sometimes referred to as over constraining the element to have zero shear at two points thus causing locking.
Numerical Examples Using Mindlin Plate Elements
We now demonstrate the concepts we have been discussing with numerical simulations using various Mindlin plate elements. We will numerically demonstrate that futl integration leads to locking that can be suppressed by selective-reduced integration. This section will present numerical examples of Mindlin plate element used on a square plate with various boundary conditions and loads. We model the square plate with lOxlO elements as shown in Figure 3.2
23
-
a
-I b
Figure 3.2 Square plate
The data for the plate consists of the following
E = 3 x 107 psi v= 0.3 a= b = 1 inch.
r
I X
Considerations similar to those in Mindlin beam elements can be applied to the Mindlin plate element. If the plate is thin, the transverse shear strain in both directions (equations (3.15)) should be small.
YzJ = w, .. l -O .. l = 0
y 31 =w, .. -O .. :::0 I I
(3.15)
Using bilinear shape function for Mindlin plate elements, two-point Guassian quadrature is required to exactly integrate the shear stiffitess. This full integration brings in the higher order terms of the rotational degrees of freedom that the terms
24
corresponding to the transverse defonnation do not have. Thus, full integration can lead to shear locking unless both transverse shear strains are evaluated at x1 = x2 = 0 (selective-reduced integration). The point being that the excess
monomials in the tenns corresponding to the rotational degrees of freedom are eliminated. Using selective integration with Mindlin plate elements can prevent locking and gives acceptable results for thin plates as shown in Figure 3.3. We can see that when the plate become thin, exact integration gives unacceptable results due to locking.
r 'I 0.6 ~ L••••<<••}\(\ .. c 0.4
0.2
a) Clamped-edges with distributed load.
25
.. 0.8 u
t I o.s u i E .. c 0.4
0.2
0 0 50 100 150 ... b) Simple supports with distributed load
0 50 100
200
150
c) Simple supports with point load at the center of plate
250 300
200 250
Figure 3.3 Numerical/ Analytical Ratio of center deflection of 1 Ox 10 element, square plate. Thin plates correspond to large ab/t.
26
Instability
Generally, finite elements do not give exact results. We have demonstrated that the use of a more accurate stiffuess integration might make an element too stiff causing locking. We saw that using selective-reduced integration eliminates locking. However, using selective-reduced integration can also cause something worse than locking called instability.
Zero Strain Energy in Mindlin beam
Instability is sometimes referred to as a zero strain energy mode. This is because the global displacement matrix results in zero strain at the Gauss points so that the mode is easily excited. For this reason, instability is often associated with rank deficiencies of the element stiffuess matrix which is caused by lower-order integration.
To explain the term "zero-energy mode", we form an expression for the element strain energy.
U. = ±{df[k]{d} = ± f {&f[D]{&}d4 A,
where [k] = j[Bf[D][B]dA A,
and {E} = [B]{d}
When we evaluate [ k] numerically, it contains only the information that can be
sensed at the sampling points of the quadrature rule. If the strains vanish at all Gauss points for a certain mode {d}, then U. must be equal to zero, and [k] must be a zero-stiffuess matrix. We know that the strain energy must vanish for only rigid body motion. If it vanishes for other motions, an instability has happened.
To support the above discussion, we consider the in-plane twisting mode of the two Mindlin beam elements shown in Figure 3. 4. We will compare the strain energy using stiffuess matrices derived with both full integration and selective-reduced integration.
27
·. ·.
·. .·
o------~----~e~---c~-----Q
.· ·.
Figure 3.4 Instability in the Mindlin beam element, dashed-line shows in-plane twisting instability mode.
The global displacement matrix of the 3 nodes is
WI
OJ
{d} = w2
82
w3
83
The displacement matrix ofthe in-plane unstable twisting mode is
{d} =
0
1
0
-1
0
28
(3.16)
(3.17)
The global stiffuess matrix, assembled from the element shear stiffuess in equation (3.8) (obtained by selective reduced integration), is
L -1
L 0 0
2 2 L L2 L L2
0 0 2 4 2 4
-1 L
2 0 -1 L
ks = kGA 2 2 (3.18) g L L L2 L2 L L2
- 0 - --2 4 2 2 4
0 0 -1 L L 2 2
0 0 L L2 L L2
- -2 4 2 4
where k; is global shear stiffuess matrix obtained by selective-reduced integration.
Now by applying equations (3 .17) and (3 .18) into the expression of strain energy, we obtain
UT = ~ {d} 1 [k;]{d}
=0
where U 1 denotes the total strain energy.
This result corresponds to the definition of zero-strain energy mode. In a like manner
we use full integration and obtain the exactly integrated shear stiffuess matrix, k {
29
L -1
L 0
2 2 L L2 L L2
0 - -2 3 2 6
-1 L
2 0 -1 kr = kGA 2
g L L L2 2L2 L - 0 --
2 6 3 2
0 0 -1 L
I 2
0 0 L L2 L
-2 6 2
Now , calculating the total strain energy, we obtain
UT = ~ {d}T[k{ ]{d}
2L2
= 3
0
0
L 2 (3 .19) L2
--6 L 2
L2 -3
We can see that the zero-strain energy mode is not present with exact integration. However we should be concerned about exact integration because of shear locking.
Eigenvalue Analysis of the Zero Strain Energy Modes
There is another approach to detect instability in the finite element method. This approach performs an eigenvalue analysis of the element stiffness matrix. Consider the element equilibrium equation [ 12]
[k]{d} = {f}
Now if the load matrix {f} is proportional to element nodal displacements { d} through a factor A., we have
[k]{d} = {f} = A.{d}
30
or ([k]-A[/]){d} = {0} (3.20)
This is an eigen problem with the eigen values of { k }" being the A. Each A; corresponds
an eigenvector { d} ;. If each {d); is normalized so that {d}; {d}, = 1, from equation
(3.20), we obtain
or 2U; =A, (3.21)
where U; is the strain energy in the element when its nodal d.o.f are normalized displacements { d} ;.
[k] is complete element stiffness matrix of the element in an infinite media.
Equation (3 .21) can be interpreted in the same way as was the expression of strain energy, that is [k] should yield A;= 0 for the rigid body motion case only. If it is not a rigid body motion and A; = 0 then instability is present. Generally a solution with an instability is easy to detect as it usually contains short wavelength, high amplitude unphysical deformations.
Analysis of Instability and Locking of Bilinear Mindlin Plate Elements Using Spatial Fourier Modes
Corresponding to the above discussion we might say that instability happens when strain energy vanishes for deformation modes not related to rigid body motion. Next we will examine the instability and locking by applying Fourier modes on an infinite media. We begin by writing the expression for strain energy for a linearly elastic isotropic Mindlin plate
31
+ --{} + --{} dx Eh [(~ )
2
(~ J2f
4(1 + v) If acl I ac2 2 I 2 (3.22)
We will use the definition above to calculate a continuous strain energy to compare with the strain energy calculated from the finite element methods. We write the nodal deflections w,, ()"'; and ()"' ; as general harmonic waves in 2 dimensions
I 1
(3.23)
For thin plates we can neglect the transverse shear deformations, so that
(3.24)
(3.25)
We note that in Mindlin theory, other modes may be present but we expect these to be the dominant modes. These three equations above can be written via nodal shape functions as
(3.26)
(3.27)
(3.28)
where w is the node element deflection () "'• is the small angle rotation in x 1 direction
()"' is the small angle rotation in x2 direction 1
32
k"' , k"' is the wave number in each direction I 2
dx1 is the horizontal length of element dx2 is the vertical length of element n is the element index in horizontal direction m is the element index in vertical direction
Equations (3.26), (3.27) and (3.28) allow us to write the displacements as a function of the wave number and then calculate the strain energy. We will compare the continuous and finite element strain energies calculated using these modes.
case 1 ) k"' = k"' = 0. I 2
This case corresponds to rigid body motion. It is certain that strain energy must vanish. This is true for any method.
case 2) maximum number of wave, k"' x, = 1r, k"' x2 = 1r I 2
The maximum admissible wave number that can be represented by nodes of finite elements for the transverse deformation is presented in Figure 3.5. This is the characteristic instability called w-hourglass mode.
.· .·
.· 0~-----c----~e~----~--------Q
·. .· .·
·.
Figure 3.5 The maximum admissible wave number.
33
The wave number corresponding to the wave length is
(3.29)
where A. is the length of one wave. At the minimum admissible wave length A, which is equal to the length of two elements, 2dx, we have the maximum wave number
7i kmax = dx (3.30)
The deformation at the maximum admissible wave number is not a rigid body motion. If we apply the maximum wave number into equations (3 .26), (3 .27) and (3 .28) and calculate the strain energy we will obtain, in some cases, the zero strain energy mode. We will show that the zero strain energy mode only appears, in some cases, at the maximum admissible wave number. We will do this calculation numerically with a single one inch square element. In the case of Mindlin plate elements, only the whourglass mode is possible. We will discuss this in more detail later. We can write the displacement
(3.31)
Using equation (3.31) in the expression for strain energy, we can calculate the strain energy as a function of the wave number using stiffness matrices obtained using full integration and selective-reduced integration. We compare these solutions with the continuous solution using the same Fourier modes. Figure 3.5 presents the strain energy of a single one inch square Mindlin plate element. We set the shortest wave length equal to 2 inches so that (k" x1)m"" = (k" x2 )m"" = 7i. Consider Figure 3.5 a
I 2
and c, The locking behavior makes the strain energy obtained from full integration too big, even at a very small wave number. In Figure 3.5 a and c we plot the logarithm of strain energy as the strain energy obtained from continuous case is very small
34
compared to that obtained using full integration. Compare these results with the results from selective-reduced integration in Figure 3.5 b and d, we can see that the smaller the wave number, the more accurate these elements are. Practical use of these elements does not generally consider deformations near the maximum admissible wave number and the accuracy of these elements would be acceptable were it not for the behavior at the maximum admissible wave number. At the maximum admissible wave number the strain energy using selective-reduced integration vanishes indicating the presence of an unstable mode.
Strain Energy Log (lb.inch)
"'- Numerical Solution 5
0 Analytical Solution
-5
-10
-15 0 0.5 1 1.5 2 2.5 3 3.5
Wave Number (kAx)
a) Strain energy from full integration and analytical from ktu = 0 to ;r
35
Strain Energy (lb.inch)
Numerical Solution
2000
1500
1000
500 Analytical Solution
0 0 0.5 1 1.5 2 2.5 3 3.5
Wave Number (k.1x)
b) Strain energy from selective-reduced integration and analytical from kta = 0 to 1r
Strain ,,---------,----------.---------.------------r---------.---------~~-----.
Energy Log (lb.inch)
1
-5
-10
-15
-20
Numerical Solution
Analytical Solution
~5~--~----~----~----~----~----~--~ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wave Number (k.1x)
c) Strain energy from full integration and analytical from kta = 0 to w'IO
36
Strain Energy (lb.inch)
0.03
0.02
0.01
0.05 0.1
Numerical Solution \
0.15 0.2 0.25 0.3 0.35 Wave Number (k!u)
d) Strain energy from selective-reduced integration and analytical from kllx = 0 to wlO
Figure 3.5 The strain energy of a single one inch square Mindlin plate element in an infinite media, thin plate limit.
We note that Figure 3.5 indicates the expected result that for a given wavelength,
2~ , the numerical results become more accurate as llx ---+ 0 .
37
Typical Instability Mode in Mindlin Plate Element
We split [k] into the bending stiffness [kb] , and the transverse shear stiffness [ks]. We integrate the bending stiffness fully and we under intergrate the shear stiffness which makes only two modes of instability possible, w-hourglass and inplane twist mode. The latter can not happen with two or more element because it can not be communicated between adjacent elements, so that only the w-hour glass mode appears. In general the global stiffness matrix is singular by itself and zero-energy modes can be prevented after boundary conditions are applied. From the assumption that the boundary conditions preclude the rigid body mode from appearing in one element, it also precludes the mode in the remainder of the mesh [8]. A critical test is to clamp one node in the middle of the square plate, and the instability appears as shown in Figure 3.6. Now we fix one node that is adjacent to the center node. The instability is gone. To retain symmetries as shown in Figure 3. 7, we fix 3 adjacent clamped nodes in the middle of the plate.
38
0 15
Figure 3.6 w-hour glass mode with clamped center node of IOxiO Mindlin
plate elements with selective integration and distributed load.
39
15
0.01
0
-0.01
-0.02
-0.03
-0.04 15
0 0
Figure 3. 7 Deflection of 3 clamped center nodes of lOxlO Mindlin plate elements with selective integration and distributed load.
40
15
CHAPTER4
THE DEVELOPMENT OF A MINDLIN PLATE ELEMENT
The preceding chapter was devoted to the study of instability and locking in Mindlin plate elements with bilinear shape functions. We showed that using exact integration of the element stiffness matrix resulted in shear locking. Shear locking can be suppressed by adopting selective-reduced integration, however for some specific situations selective-reduced integration introduces instability. In this chapter we will present an reliable element that does not lock and contains no unstable modes[4].
Approach
The development of Mindlin plate element obtained from the method of selectivereduced integration will be tried with a different point of view. Let us consider the curvature of Mindlin plate element.
(4.1)
(} -w .1'1 'XI
The first three terms in the curvature matrix are bending stress-strain relations, a 11 ,
0"22 and 0"12. respectively. The last two terms on the bottom are the transverse shear deformation. For the isoparametric element we used the standard nodal shape interpolations as follow,
(4.2a)
41
1 N3 = - (1+~(1+7])
4
Where each node is located in Figure 4.2
( -1,1) .--------....._,(1, 1)
(-1,-1) (1,-1)
Figure 4.1 The quadrilateral element in~. 77 coordinate (the master element)
(4.2b)
(4.2c)
(4.2d)
In the previous chapter we showed that selective-reduced integration solves the shear locking problem and gives good convergence behavior, but leads to unstable modes.
To motivate the development of a better Mindlin plate element we consider the positive characteristics of selective-reduced integration. The one Gauss point located at the center of an element results in constant transverse shear strains over the element in each direction. The element presented in this chapter has similar properties in that one of the shear strains is constant along element edges. The shear strain is obtained by approximating the slope of the transverse deformation with out strict adherence to the element shape functions. The resulting expression allows us to use two Gauss points and integrate the resulting polynomials exactly.
42
From the definition oftransverse shear strain
(4.3)
(4.4)
We can define these transverse shear strains with different interpolation functions from the in plane stress-strain terms [ 13]
(4.5)
(4.6)
where Yl3 and ~I are defined in Figure 4.2, and r~J' r~J' r:l and r~l are shear strains at points B, D, A, C respectively
/3
; ,., 4 c 3
r 1 B D
A L ~ 1 IE-~--- /1------i!J 2
Figure 4.2 Convention for the transverse shear deformation
43
Approximating the transverse shear strain from equations (4.3) and (4.4) and using equations (4.5) and (4.6) we obtain
(4.7)
(4.8)
Unlike the shear strains resulting from strict observance to the element shape functions these shear strains may have infinitesimally small values at two points and thus will not lock when full integration is used. We are now ready to modify the stiffness matrix by using these difference interpolation functions with linear shape functions, the curvature can be modified to be
(4.9)
Where Y2J and YJJ are defined in equations (4.7) and (4.8). The [Bs] matrix can be straight forwardly constructed as
44
(1-;) (1- T/)
/4 /I
0 (1- T/)
2 (1-;)
0 2
(1 +;) (1- T/)
/2 /I
0 (1- T/)
2 (1-;)
0
[BsY = ~ -
2 (I+;) (1 + T/)
/2 /3
0 (1 + T/)
2 (1+;)
0 2
(1-;) (1 + T/)
/4 /3
0 (I+ T/) (4.10)
2 (1-;)
0 2
Then from the shear stiffness matrix is calculated
[ks] = j[Bsf[DM ][Bs]d4 A
Using this new mixed interpolation function, Mindlin plate elements do not lock and because every integration is done exactly, it is certain that there are no unstable modes. It should be noted that these mixed interpolation functions may be under integrated as well as with the bilinear interpolation function but we must beware of the possible unstable modes. For example, if this new mixed interpolation function is under integrated and applied to the clamped-center node problem shown at the end if chapter 3 also leads to unstable modes. However, we can use full integration to solve
45
this problem without locking or instability and this will be demonstrated by using Fourier modes to calculate the strain energy.
Numerical Examples
We used the same material and dimensions presented before but with the mixed interpolation with full integration discussed above. Note that the locking behavior exhibited with full integration in the previous chapter does not appear here.
"i
i "i c: • ::a • u "i E :!1 c:
Simple a~pport-dillributed load
1.12
1.1
108
106 Bilinear: Selective-reduced integration
1.04
1.02 Mix interpolation:Full integration
0.98 +-------+----~,__-----+--------t-----.-------<
0 50 100 150
8/t
200 250 300
a) simple support-distribute load
46
1.18
1.16
1.14
11.12
E 1.05 :II c: 1.04
1.02
0.98
Simple Support-point load
ablt
b) simple support-point load
Clamped edgea diatributld load
1.1
108
1.05
I 1.04
l! :!: 1.02 • I :II c:
0.98
0.96
0.94
ablt
c) clamp-distribute load
Figure 4.3 NumericaV Analytical Ratio of center deflection of 1 Ox 10 element, square plate. Thin plates correspond to large ab/t.
47
The Accuracy of the Mixed Interpolation Element
The preceding chapter, we used strain energy calculations to analyze the accuracy of selective-reduced integration. We will use the same to analyze the mixed interpolation element discussed in this chapter. The purpose of the element introduced in this chapter is to prevent both instability and locking by adopting a different nodal interpolation function with full integration of both the bending stiffness matrix and the shear stiffness matrix. Considering Figures 4.4 a and b we can see that for small wave numbers the element produces acceptable results. There is no shear locking and no zero-strain energy mode.
Strain 15 Energy Log(lb.inch) 1 0
5
0 Analytical Solution
-5
-10
-15 0 0.5 1 1.5 2 2.5 3
Wave Number (kAx)
a) Strain energy from mixed interpolation and analytical from k!lX = 0 to ;r
48
3.5
Strain Energy (lb.inch)
0.05 ..--~---r---..--~---.----.---r----.
0.04
0.03 Numerical Solution ----+1
0.02
0.01
OL---~-----=~~----~--~----~--~ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wave Number (kAx)
b) Strain energy from mixed interpolation and analytical from kllx = 0 to w'l 0
Figure 4.4 The strain energy of a single one inch square Mindlin plate element in an infinite media, thin plate limit.
49
CHAPTERS
CONCLUSION
A computer model was developed to simulate the bending action of Mindlin plate elements. This program was also dedicated to the study of shear locking, instability and strain energy in thin plate deformations. The selective-reduced integration with standard bilinear shape functions is introduced to eliminate shear locking successfully. This technique requires simple programming and produces a very good solution for thin plates (more than 99% accurate compared to classical analytical solutions). Unfortunately, this selective-reduced integration leads to unstable modes in the finite element solution. These unstable modes can be precluded via an appropriate boundary condition.
Finally, by considering the plate as an extension of two-node beam elements to two-dimensions, the different interpolation functions for transverse shear deformation can be constructed. The mixed interpolation produced even better convergence results than the selective-reduced integration method did. This mixed interpolation function does not lock and also does not contain any unstable modes. It is obtained by a slight modification of the shear stiffness matrix for an element in cartesian coordinates. However, this element is quite complicated in general geometries element compared to the previous techniques. This is because it involves tensorial shear strains in arbitrary directions.
We presented two different viable element formulations, selective-reduced integration and mixed interpolation. Each has practical advantages and disadvantages. The instability from selective-reduced integration results from use of boundary conditions not usually seen in practical problems. The mixed interpolation formulation for arbitrary quadrilateral shapes can be cumbersome. We let the reader decide which technique is appropriate to their problems with optimization of computer work and the worker's time.
50
APPENDIX A
Stress Resultants in a Flat Plate
Consider Figure A. 1 by assuming that the material is isotropic. Let M11 be the bending moment per unit length of x2 axis. We obtain [6]
h
2
MII = J xJaiidxJ h --2
and let M 12 be the twisting moment M12 per unit length of x2 axis, we get
h
2
Ml2 = f XJU 12dx3 h --2
Similar development in x1 axis, we obtain
and
h
2
M22 = J X3U22dx3 h
h
2
--2
M21 = J X3a12dx3 = M12 h --2
note: positive direction is indicated by the right-hand rule for moments.
(A. I)
(A.2)
(A.J)
(A.4)
If we let Q1 and Q2 be the transverse shears per unit length in X2 axis and X1
direction. We can find equations for transverse shear forces per unit length.
51
(A.5)
--2
h
and (A.6)
--2
Resultant moments and transverse shear forces are shown in Figure A.2
Next, is the derivation of the equation for the traction force on a cross sectional surface of the element. Let N11 be the traction force of the element per unit length in the x2 direction on the middle surface. From Figure A. I we get
q l
I. Figure A.l Stresses that act on cross sections of a differential element of a homogeneous linearly elastic plate. q(x.,x2) is the distributed force per unit area on lateral surface
52
h
2
Nu = J au~3 h
2
h
--2
h
2
Nl2 = J 0"12~3 h --2
Traction forces are shown in Figure A.3
• ~+~
! ~~& 0
q
Figure A.2 Resultant moments and shears
53
(A.7)
(A.8)
(A.9)
----+ X2
Q2
0--+~
+ Mn f M22
)I
Figure A.3 Resultant tractions
Nu )I
Normal stresses a,, and Oi2 vary linearly with X3, are maximum at ±h/2 and zero at x3 = 0, and are associated with bending moments M 11 and M22· The shear stress 0"12
also varies linearly with x3 and is associated with twisting moment Mt2· Transverse shear stresses o-13 and Oi3 vary quadratically with X3. Integration of equations (A.l)(A.6) results in
(A.IO)
(A. II)
54
The maximum magnitude of a23 and a 13 is at x3 = 0
1.5Q2 (723 =-
t
Force and Moment Balance of Flat Plates
(A.12)
(A.l3}
(A.l4}
We will start the derivation of the force and moment balance equations for a flat plate by considering the equations of motion
(A.l5}
(A.l6}
(A.l7)
Force Balance for Plates
by using equation (A.15) divided by thickness h and integrating with respect to x3
[14]
we obtain
h
_!_ f (A.l)dr3 h h
--2 --
2
55
(A. IS)
A parallel development using equation (A.l6) results in
(A.I9)
Equations (A. IS) and (A.I9) are force balance equations for the plate element per unit thickness as shown in Figure A.3 . X1 and X2 are the force per unit area of loading on top and bottom of the plate in the x1 and x2 direction, respectively.
Moment Balance for Plates
We now consider
(A.20)
--2
integrating equation (A.20) yields
(A.21)
Equation (A.21) is the vertical force balance for a plate element, as shown in Figure A.2
We now form
h
2
J x 3 (A .15)dt-3 and obtain h --2
(A.22)
56
h
2
Similarly we form J x3 (A.16)dx3 to obtain h --2
(A.23)
Equations (A.22) and (A.23) are the balance of moments about the x1 and x2 axes, respectively. The physical picture is shown in figure A.2.
57
APPENDIX B
Derivation of the Four-Node Quadrilateral Mindlin Plate Element
(-1,1)..-------~(1,1)
(-1,-1) (1,-1)
Figure B. I The quadrilateral element in~, 71 coordinates (the master element)
We first develop the standard, bilinear shape function on the master element (Figure B.1) which is defined in 71, ~ coordinates. The shape functions are defined such that N; is equal to unity at node i and equal to zero at other nodes. The four shape functions can be written as
(B. Ia)
(B.1b)
(B.1 c)
58
or I M= -(t-;;,)(I+7777,) 4
where ;, and 71, are the coordinates of node i
(B.ld)
(B.Ie)
We now express the displacement field within the element in terms of the nodal values.
4
w=I:N1w1 i=l
4
(}x, = LN;8xJ i=l
4
(}x, = LN18~i r=l
which can be written in matrix from as
{u}=[N]{d}
where
59
(B.2a)
(B.2b)
(B.2c)
(B.3)
WI
(} :r I I
(}:rl 1
{d} = (B.4)
wn
(} :r,n
(} :r n 1
and
[N' 0 0 N2 0 0 NJ 0 0 N4 0
j.J N= ~ Nl 0 0 N2 0 0 NJ 0 0 N4 (B.S)
0 Nl 0 0 N2 0 0 NJ 0 0
Next, we express the derivatives of a function of x, in ~. Tf coordinates, using the chain rule,
(B.6)
or (B.7)
60
where J is the Jacobian matrix
From equations (B. I) and (B.2) ,we obtain
where
from equation (1.27), the curvature is defined as
61
by applying equation (B. 7) we obtain, after inversion
(B.S)
(B.9)
(B.IO)
62
from equations (B.8), (B.9) and (8.10), we get {K} in the form
0 0 J22 -J12 0
0 0 0 0 -J21 1
{K}=- 0 0 J21 J11 J22 detJ
J21 -Jll 0 0 0
-J22 J12 0 0 0
Now from the interpolation equation, we have
cw 0~ cw a,
O{;kl
0~ O{;kl - =[G]{d} a, iJ(}x2
0~ iJ(}x2
a, Brl
6r2
63
0 0 0
Jll 0 0
-J12 0 0
0 0 detJ
0 detJ 0
cw -0~ cw -a,
iJ8 ....
0~ iJ(} ....
(B. II) a, iJ(} ... 2
0~ iJ(} ... 2
a, e .... () ... 2
(B.12)
where
-(1-'7) -(1-~ 0 0 0 0 0 0
0 0 -(1- '7) -(1-;) 0 0 (l-~(l-'7) 0
0 0 0 0 -(1-'7) -(1-~ 0 (1- ~(1- '7)
(1- '7) -(1+~ 0 0 0 0 0 0
0 0 (1- '7) -(1 +;-} 0 0 (1+~(1-'7) 0
(Gf = .!_ 0 0 0 0 (1- '7) -(1+~ 0 (1+~(1-'7)
4 (1 + '7) (1+~ 0 0 0 0 0 0
0 0 (1 + '7) (1+~ 0 0 (1 +~(1 + '7) 0
0 0 0 0 (1 + '7) (l+~ 0 (1+~(1+ '7)
-(1 + '7) (1-~ 0 0 0 0 0 0
0 0 -(1 + '7) (1-~ 0 0 (1- ~(1+ '7) 0
0 0 0 0 -(1 + '7) (1-~ 0 (1- ~(1+ '7)
equations (B. II) and (B.12) yields
[B] = [A][G] where
0 0 122 -112 0 0 0 0
0 0 0 0 -121 111 0 0
[A]- 1 0 0 121 111 122 -112 0 0 - det1
121 -111 0 0 0 0 0 det1
-122 112 0 0 0 0 det1 0
Finally we can calculate the element stiffness from
[k] = j[Bf[DM ][B]dA A
64
Numerical Integration
Consider the problem of numerically evaluating a one-dimensional integral of the form [11]
I
I= f f(f)d; -I
The Gaussian quadrature approach for evaluating I with n-point approximation is given below
where and
I
I= J f(f)d; ~ wJ~(;I) +w2J(;2)+ ...... +wnJ(;n) -I
w1, w2, ..... ,wn are the weights ;1, ;2, ..... ,;n are Gauss points
(B.13)
From equation (B.13), n-point Guassian quadrature provides an exact answer for polynomial of order (2n-1) or less.
Two Dimensional Integrals
We can write the equation to evaluate the two-dimensional integrals of the form
I 1
I= f f J(;, q)dgiq (B.14) -1-1
Now, from equation (B.13)
n n
~ LLw;w1J(;;,T]1 ) (B.15) i=lj=l
65
Stiffness Integration
Introduction of [B] into the expression for the stiffness matrix, we may now numerically evaluate the element stiffness matrix [II ] .
172 = 0.57735 0 0
__..
1]1 = -0.57735 0 0
;I= -0.57735 ;2 = 0.57735
Figure B.2 two-dimensional Gaussian quadrature with 2x2 rule.
I I
[k]= J f[Bf[DM][B]detJd;dq (B.I6) -1-1
where [B] and detJ are the functions of ; and 7].
Let ¢J(;,q) = [Bf[D][B]detJ (B.I7)
Then, If we use a 2-point Gaussian rule in each direction, we get
66
where
[k .] = WJ2rp(;l, 771) +WI w2;(;1, 772) + Wz wlrp(;2, 771) + w; ;(;2, 772) (B.18)
;I = 711 = - YJJ ;2 = T/2 = y../3
and the Gauss points are shown above diagrammatically
67
REFERENCES
1. T. J. L. Hughes, R. L. Taylor, and Kanoknukulchai, "A Simple and Efficient Element for Plate Bending ," International Journal for Numerical Methods in Engineering, 11, no. 10(1977), 1529-1543.
2. R. H. Macneal, "A Simple Quadrilateral Shell Element," Computers and Structures, 8(1978), 175-183.
3. T. J. R. Hughes, "Generalization of Selective Integration Procedures to Anisotropic and Nonlinear Media, International Journal of Numerical Methods in Engineering, Vol. 15, 1980, 1413-1418
4. T. J. R. Hughes and T. E. Tezduyar, "Finite Elements Based Upon Mindlin Plate Theory With Particular Reference to the Four-node Bilinear Isoparametric Element," Journal of Applied Mechanics, (September 1981 ), 587-596.
5. K. C. Park and D. L. Flaggs, "A Fourier Analysis of Spurious Mechanisms and Locking in The Finite Element Method," Computer Methods in Applied Mechanics and Engineering, 46 ( 1984 ), 65-81.
6. A. B. Boresi, 0. M. Sidebottom, F. B. Seeley and J. Smith, Advanced Mechanics of Materials, 3rd ed., New York: John Wiley & Son Inc., 1985.
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