-CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 401

SE F N -C NF FINI E ELE E

F PL ES

UllU-ENG-73 .. 2019

RING DES I AN [SIS

SHELLS

By CHANG-KOON CHOI

W. C. SCHNOBRICH

Issued as

a Technical Report

of a Research Program

Sponsored by

THE OFFICE OF NAVAL RESEARCH

DEPARTMENT OF THE NAVY

Contract NOO0l4-67-A-0305-00l0

UNIVERSITY OF ILLINOIS

at URBANA-CHAMPAIGN

URBANA, ILLINOIS

SEPTEMBER 1973

UIlU-ENG ... 73",,2019

USE OF NONCONFORMING MODES IN FINITE ELEMENT

ANALYSIS OF PLATES AND SHELLS

By

Chang-Koon Choi

and

William C. Schnobrich

Issued as a Technical Report of a Research

Program Sponsored

by

The Office of Naval Research Department of the Navy

Contract N 00014-67-A-0305-00l0

University of Illinois Urbana, Illinois

September, 1973

iii

ACKNOWLEDGMENT

This report was prepared as a doctoral dissertation by Mr. Chang­

Koon Choi submitted to the Graduate College, University of Illinois at Urbana­

Champaign, in partial fulfillment of the requirements for the degree of Doctor

of Philosophy in Civil Engineering.

The study was conducted under the supervision of Dr. W. C. Schnobrich,

Professor of Civil Engineering, as part of a research study of numerical analysis

supported by the Office of Naval Research under contract N00014-67-A-0305-00l0.

Also the support of Mr. Choi's graduate study by a University of Illinois Fellow­

ship is gratefully acknowledged.

The numerical calculations were performed on the IBM 360/75 system

of the Computing Service Office of the University of Illinois.

CHAPTER

2

3

iv

TABLE OF CONTENTS

INTRODUCTION .

1 . 1 Genera 1 . 1.2 Literature Survey. 1.3 Object and Scope. 1.4 Notations

ISOPARAMETRIC ELEMENTS.

Page

1 2

10 11

15

2.1 General. 15 2.2 Isoparametric Shell Element 16 2.3 Ahmad-Irons Modification 17 2.4 Reduced Integration Technique. 20 2.5 Addition of Nonconforming Displacement Modes. 22

ELEMENT FORMULATION.

3.1 Variational Principle 3.2 The Basic Geometry and Displacement Field. 3.3 Nonconforming Displacement Modes. 3.4 Strain-Displacement Relationships 3.5 Stress-Strain Relationships 3.6 Element Stiffness Matrix 3.7 Selection of Nonconforming Modes. 3.8 Forced Interelement Boundary Compatibility

26

26 28 29 31 33 34 37 39

4 ASSEMBLY AND SOLUTION OF THE LOAD­DISPLACEMENT EQUATIONS. 42

5

4.1 General. 4.2 Assemblage of Stiffness Matrix and

Boundary Conditions . 4.3 Load Vector. 4.4 Solutions of Load-Deflection Equations. 4.5 Strains. 4.6 Stresses and Forces .

NUMERICAL RESULTS

5.1 General . 5.2 Deep Beams . 5.3 Plates 5.4 Cylindrical Shell. 5.5 Elliptic Paraboloid.

42

42 43 44 45 45

47

47 48 51 54 56

6

v

5.6 Spherical Shell . 5.7 Hyperbolic Paraboloid. 5.8 Conoidal Shell

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDIES .

6.1 Conclusions 6.2 Suggestions for Further Studies.

LIST OF REFERENCES .

APPENDIX

A SHAPE FUNCTIONS.

B LOCAL COORDINATE SYSTEM .

C LOCAL STRAINS AND TRANSFORMATIONS.

D NUMERICAL INTEGRATIONS .

E TYPICAL COMPONENTS OF ELEMENT STIFFNESS MATRIX

Page

57 58 59

62

62 63

65

103

105

107

110

. 112

Table

2

3

vi

LIST OF TABLES

Element Designations.

Comparison of Results of Cantilevered Beam Using Various Elements

Abscissae and Weight Coefficients of the Gaussian Quadrature Formula.

Page

70

71

72

Figure

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

vii

LIST OF FIGURES

3-DIMENSIONAL ISOPARAMETRIC ELEMENTS .

LINEAR ISOPARAMETRIC ELEMENT UNDER PURE BENDING

RESTORING THE ACTUAL BEHAVIOR OF THE ELEMENT BY ADDING NONCONFORMING MODE.

DEFINITIONS OF GLOBAL AND LOCAL COORDINATES.

NONCONFORMING DISPLACEMENT (w) MODES FOR LINEAR ELEMENT

NONCONFORMING DISPLACEMENT (w) MODES FOR QUADRATIC ELEMENT

INTEGRATION POINTS FOR PLATE (n = 3) .

GAUSS INTEGRATIONS .

TYPICAL UNIFORM MESH (2x2) OF CLAMPED SQUARE PLATE UNDER CONCENTRATED LOAD AT THE CENTER.

CENTRAL DEFLECTION OF CLAMPED SQUARE PLATE UNDER CONCENTRATED LOAD AT THE CENTER.

CENTRAL DEFLECTION OF CLAMPED SQUARE PLATE .

COMPARISON OF CENTRAL DEFLECTION FROM VARIOUS PLATE BENDING ELEMENTS.

CLAMPED EDGE CIRCULAR PLATE UNDER UNIFORM LOAD.

GEOMETRY AND MATERIAL PROPERTIES OF CIRCULAR CYLINDRICAL SHELL

DISPLACEMENTS OF CIRCULAR CYLINDRICAL SHELL UNDER UNIFORM LOAD .

STRESS RESULTANTS OF CIRCULAR CYLINDRICAL SHELL UNDER UNIFORM LOAD .

GEOMETRY AND MATERIAL PROPERTIES OF A SQUARE ELLIPTIC PARABOLOID

Page

73

74

75

76

77

78

79

79

80

80

81

82

83

84

84

86

89

viii

Page

18 VERTICAL DISPLACEMENT OF SIMPLY SUPPORTED SQUARE ELLIPTIC PARABOLOID UNDER UNIFORM LOAD . 89

19 STRESS RESULTANTS OF SIMPLY SUPPORTED SQUARE ELLIPTIC PARABOLOID UNDER UNIFORM LOAD . 90

20 GEOMETRY AND MATERIAL PROPERTIES OF SPHERICAL SHELL. 91

21 STRESS RESULTANTS OF SPHERICAL SHELL UNDER UNIFORM LOAD. 92

22 GEOMETRY AND MATERIAL PROPERTIES OF HYPERBOLIC PARABOLOID 93

23 VERTICAL DEFLECTION ACROSS MIDSPAN OF CLAMPED HYPERBOLIC PARABOLOID UNDER UNIFORM NORMAL LOAD . 93

24 STRESS RESULTANTS OF CLAMPED HYPERBOLIC PARABOLOID UNDER UNIFORM NORMAL LOAD 94

25 GEOMETRY AND MATERIAL PROPERTIES OF CONOIDAL SHELL . 95

26 DISPLACEMENTS OF SIMPLY SUPPORTED CONOIDAL SHELL UNDER UNIFORM LOAD. 95

27 STRESS RESULTANTS OF SIMPLY SUPPORTED CONOIDAL SHELL UNDER UNIFORM LOAD. 97

28 DEFLECTIONS OF CLAMPED CONOIDAL SHELL UNDER UNIFORM LOAD. 100

29 STRESS RESULTANTS OF CLAMPED CONOIDAL SHELL UNDER UNIFORM PRESSURE 101

Chapter 1

INTRODUCTION

1.1 General

Various types of shell structures have been built for various pur­

poses in a number of countries in the past. With the structural efficiency

of this type of structure, it is obvious that more such structures will be

built in the future. Architectural or roof shells are one form of this

structural shape which has attracted a good deal of engineering research ef­

fort and designers· attentions as well. Aesthetics, long column-free span­

ning capacities and economy, both in the sense of a weight-to-space ratio

as well as in material quantities used, have made this type of structure very

popular.

In recent days, however, it has been observed that while architec­

tural or roof shell construction has somewhat tapered off, particularly in

countries where the wages of the construction worker have risen sharply,

other types of shells, such as aeronautical structures, nuclear reactor

vessels and marine structures, have become more widely used. Consequently,

recent research efforts have been directed in these latter areas. The shells

to be analyzed have also become more complicated.

Solutions by classical methods of shell analysis require the solving

of complicated governing differential equations. Therefore, such solutions are

extremely rare and limited in scope. Exact analytical solutions have been

possible only for those shells of simple regular geometry, ideal boundary

conditions and simple loadings. This is in great measure due to mathematical

2

complexity of the resulting equations. Simplification of the analysis pro­

cedures, such as membrane theory, although of great value, cannot provide the

detail necessary. Experimental studies as an alternative are either not widely

accepted or are too expensive and time consuming to be used for other than

research studies into a specific configuration.

The finite element method provides a powerful tool for shell analysis

as it does for many other structural analysis problems. The structure may be

approximated by an assemblage of a finite number of discrete elements inter­

connected at a finite number of nodal points. Then, piecewise continuous

displacement and/or stress fields are assumed in each element. The applica­

tion of variational principles yield a system of simultaneous algebraic equa­

tions [lJ. Once the final matrix equations have been established, the solutions

are straight-forward. Thus the unknown variables, displacements and/or stress­

es, are obtained.

It is possible to obtain numerous different finite element models

by introducing different equilibrium or compatibilit~ conditions at the inter­

element boundaries and modifying these conditions by the use of Lagrangian

multipliers [2J. According to the variables assumed in the element, the re­

sulting finite element models may be classified as 1) DispJacement (Compatible)

Models, 2) Force (Equilibrium) Models, 3) Mixed Models and 4) Hybrid Models.

A brief discussion of these alternative models will be presented later.

1.2 Literature Survey

The development and the application of the finite element method

followed, in general, in the order of plane stress problems, plate bending

3

problems and curved shell problems. Key and Krieg [3J summarized the early

stages of finite element development by different groups, i .e., mathematicians~

engineers and physicists.

Courant [4J has been frequently acknowledged as one of the first to

apply the finite element method, as known today, to structural mechanics prob­

lems. He used this method in analyzing the St. Venant torsion problem of a

prismatic bar by the means of an assemblage of triangular elements.

The first formulations of the finite element method to be used in

structural analysis were of the "Direct Stiffness Matrix·· type approach as

applied to in-plane stress problems [5,6J. This approach has the conceptual

limitation of applying to the simple elements. Later, Melosh [7J extended the

definition of the direct stiffness method to make it a variationally based

approach, expressing the finite element formulation in terms of the Minimum

Potential Energy Principle. Thus, the variational method, as a generalized

form of the Ritz method, began to be most frequently u~ed as the basis for

the element formulation.

Although the majority of the early efforts using the finite element

method belong to the classification of the displacement model, several other

forms or variations of the variational method with various displacement and/or

stress assumptions are possible. A goodly number of these have been investi -

gated. After the successes with plane stress problems [5,6,8J, plate bending

applications began to appear.

Before proceeding further with the discussion of plate bending prob-

lems, the convergence requirements for the displacement model are discussed

briefly. If the following conditions are met, the results of finite element

will converge toward the true solutions as the mesh size is reduced:

4

1. Interelement boundary compatibility,

2. Inclusion of rigid body modes,

3. Inclusion of all pertinent constant strain states.

The compatibility requirement for plate bending problems becomes

more demanding than for plane stress problems since both normal and rotational

displacements compatibilities should be satisfied. A detailed discussion of

compatibility requirements can be found in Reference 20. Melosh [9J and

Zienkiewicz [lOJ used 12 term polynomials to represent the normal displace­

ment for rectangle and quadrilateral element shapes, respectively. In both

cases normal displacement compatibility is satisfied but not the rotational

or normal slope compatibility.

It has been proved to be difficult to ~chieve intere1ement compati­

bility when using deflection shapes for the element established from conven­

tional displacement components defined only at vertices. This difficulty is

due in part to the disparity between the nUUlber of terms in a complete poly­

nomial used to define the displacements and the number of terms needed to

express the displacement field by variables defined at the vertices. Dropping

some terms from a complete polynomial may result in geometric anisotropy and

even destroy convergence [llJ.

One of the alternatives is the adoption of smooth surface interpo­

lation functions. Bogner, Fox and Schmit [12J used complete third order

Hermitian polynomial interpolation formulas for the rectangular plate ..

Fraeijs de Veubeke [13J achieved the conforming quadrilateral element by

subdividing it into 4 triangles, assuming a complete cubic polynomial dis­

placement field in each triangle and finally condensing out the interior

degrees of freedom.

5

For triangular elements the task of achieving interelement compati­

bility was found to be even more difficult. The use of higher order poly­

nomial representations of displacement functions and its derivatives is one

solution. In this case, however, additional nodes placed along the sides

and/or reference points located inside of the elements have to be utilized.

For the triangle it is impossible to achieve complete compatibility with any­

thing lower than a complete quintic (21 terms) representation. Bell [14J and

Argyris [15J adopted the complete quintic representation with the associated

21 degrees of freedom (normal displacement, and its first and second deriva­

tives at each vertex and angular displacement at each mid-side point). Bell

[14J later eliminated the mid-side node to reduce the element to one with

18 degrees of freedom.

Argyris [15~16J adopted even higher order representations, i.e.,

sextic (28 term) and septic (36 term) representations. It is obvious that

the higher the order of the representation of the displacement that is used,

the more costly the solutions become. Thus while some were adhering to

achieving full interelement compatibility by using higher order polynomials,

others sought an adequate low order incompatible element [8,17J.

The use of a cubic polynomial coupled with the concept of area

coordinates, which simplifies the development and is effective in avoiding

geometric anisotropy, has been successfully employed by Bazeley [17J and

others [18J even though it is an incomplete polynomial. The approach using

subdomain triangles was used by Clough and Tocher G19J and Clough and Felippa

[20J. The resulting elements are 9 D.D.F. and 12 D.D.F. compatible elements,

respectively.

6

Thus far, the discussion has been limited to formulations within

the scope of a minimum potential energy principle or displacement models.

Fraeijs de Veubeke [21J worked on one of the early stress models basing his

work on a minimum complementary energy principle. He showed that upper and

lower bound convergences on the solutions are obtained by the stress and by

the displacement models, respectively.

The mixed model which involves simultaneous assumptions on dis­

placements and stresses for each element was used first by Herrman [22J.

The compatibility and equilibrium are easily achieved by expanding the vari­

ables into relatively low order polynomials. Herrman used linear variations

for the triangle. With this assumption, however, the results do not appear

to demonstrate any marked superiority over other models. This mediocre be­

havior seems to be due to too simple an assumed variation of the selected

variables. Adoption of a higher order representation of variables by Visser

[23J showed some improvement in the solutions.

Herrman [22,24J did not take Reissner1s Principle as a starting

point of formulation, but later works in mixed model by Prato [25J, Pian and

Tong [2J and Will and Connor [27J are based on Reissner1s Principle. There

are several versions of element methods based on Reissner1s principle depend­

ing on the selection of the continuity requirements to be imposed along the

boundary [28J.

Formulations utilizing the hYbrid model, a terminology introduced

by Pian [26J, are based on either 1) assumed equilibrating stress fields in­

side of each element and compatible displacements along the interelement

boundary (hybrid stress model) or 2) assumed continuous displacements inside

of each element with equilibrating surface traction along the interelement

7

boundary (hybrid displacement model). Again, there may be several modified

versions of these models. The detailed classification of the variations of

this model appears in Ref. 28.

Tong [29J pointed out the advantages of the hybrid displacement

model over the hybrid stress model. In a hybrid displacement model the as­

sumed displacements need not be compatible at the beginning. The compatibil­

ity along the boundary can be produced by introducing a Lagrangian multiplier

which is physically interpreted as the self-equilibrating boundary tractions.

Harvey and Kelsey [30J followed this approach for flat triangular elements.

The first, and perhaps simplest, finite element approach applied

in shell analysis is the approximation of the surface by an assemblage of

flat elements. This can be done relatively easily by combining the uncoupled

membrane and plate bending behavior through coordinate transformations as

necessary. In this way virtually all of the plate bending elements, in proper

combination with plane stress elements, can be used in shell analysis. Green

et al. [31J first used this approach in shell analysis. Johnson and Clough

[32J and many others successfully applied flat elements in shell analysis.

Chu and Schnobrich [18J used flat triangular plate elements with higher order

displacement functions, resulting in an element with 27 generalized displace­

ments, in an attempt to provide compatibility between shell and eccentrically

placed supporting beams.

It is noted here that even with the discrete errors caused by ap­

proximating the curved shell surface by flat elements and the lack of inter­

element compatibility of some displacement components, in particular, the

discontinuity of slopes, the flat element solutions, generally, converged

to the real solutions within a reasonable limit.

8

Finally curved shell elements began to appear in finite element

shell analysis in the late 60's. Some of the difficulties experienced in

curved shell formulations are: 1) the inplane and bending actions are now

coupled and can no longer be calculated separately, 2) the continuity of dis­

placements becomes more difficult to achieve because this now requires the

three displacement components (u, v, w) to be of equal order when the dis­

placements are expressed in the global coordinate system, 3) the inclusion

of rigid body modes is much more difficult to achieve, especially when the

displacement is described in curvilinear coordinates.

Connor and Brebbia [33J expressed the radial displacement (w) by

a 12 term polynomial and the membrane displacements (u, v) with linear ex­

pansions for a shallow shell element whose geometry was defined by a quadratic

function of the coordinates. For this element, the rigid body modes are not

included nor is the compatibility completely satisfied due to the disparity

in the order of polynomials used for radial and membrane displacements.

Pecknold and Schnobrich [34J enforced rigid body motion in one of their

curved elements by introducing additional unknowns and writing the additional

constraint equations.

Bogner et al [35J used 16 term polynomial for all tnree displace-

ment components resulting in 48 D.O.F. circular cylindrical shell element.

Rigid body modes are not included in this element so it must be used only

with small arc dimensions. Compatibility requirements are however satisfied.

Cantin and Clough [36J developed a curved cylindrical element including rigid

body motion. Utku [37J achieved a relatively simple algebraic representation

for curved triangular elements by using linear variations for the three

9

displacement and two angular displacement components. Argyris and Scharpf

[16J proposed higher order variations, i.e., complete quintic polynomial for

all three displacement fields.

While the efforts to achieve complete satisfaction of convergence

conditions, as discussed earlier, have continued, it has been shown that even

with elements which do not satisfy the above conditions, the satisfactory

results can be obtained and the justification of using higher order formula­

tions essential to achieving the strict convergence conditions must be ques­

tioned.

Bazeley et al. [17J dramatized the above question by showing that

a nonconforming triangular element can produce superior results over conform­

ing elements in plate bending. Pecknold and Schobrich [34J also found that

the conforming and nonconforming elements used in investigating skewed shallow

shell problems produced almost exactly the same numerical results. The non­

essentialness of inclusion of complete rigid body modes has also been reported

in Ref, 38. It has already been shown that many of the previously discussed

elements [8,18,33J produced satisfactory convergence characteristics with one

or more of the conditions relaxed.

Recently the isoparametric element and its modifications for shell

analysis have received much attention. Ever since the first introduction of

the isoparametric element [39J, many attempts have been made to improve the

bas i c accuracy of ttli s element. Among the illOi'"e s i gni fi cant deve 1 opments in

the use of the isoparametric element for shell analysis are the modification

of a three dimensional isoparametric element to ordinary shell element by Ahmad

and Irons [40J, the reduced integration technique by Too and Irons [41J,

and addition of nonconforming displacement modes by Wilson et al. [42J.

10

The latest development in the finite element field is the extension

of the linear analysis to nonlinear problems. The major areas of interests

include elastoplastic analysis [43J, cracking problems [44J, combination of

material and geometric nonlinearities [45J and even dynamic response of non­

linear structures [46J.

1.3 Object and Scope

The behavior of the isoparametric element for three-dimensional

problems is well established and documented and the efficiency of this type

of element is also well known. The application of this element to thin shell

problems, however, is quite limited. Only the cubic elements produce reason-

able accuracy. Due to the many degrees-of-freedom involved, the economy of

the use of this element is questionable. There still exists the need for

developing a new economical shell element based on the isoparametric element.

The main objective of this study is to develop such a versatile,

yet economical, shell element based mainly on isoparametric concepts. The

weaknesses which the original isoparametric element has are corrected by

means of the addition of nonconforming displacement modes. The most beneficial

combinations of these additional modes are investigated for various orders of

elements, i.e., linear, quadratic, and cubic elements.

A computer ,program which is capable of handling problems of various

order elements, various shapes of elements and various types of shells, is

developed.

Some selected numerical results for plate and shell problems are

presented and compared with those from classical solutions, from other well

established fin~te element models and from experimental studies, whenever

11

available, in order to evaluate the efficiency of the elements presented in

this study.

The major findings and the suggestions for future studies based on

the findings of this study are presented at the conclusion of this study.

1.4 Notations

The symbols used in this study are defined where they first appear.

Following is a summary of the symbols which are used frequently.

A

B, B

c

I

0, 0

E

e

F

i ,j

J, I J I

area of an element

matrix relating nodal displacement and strains, con­

forming and nonconforming part, respectively

constraint matrix

material property matrix in global and local coordinate

system

Young's modulus

superscript indicating lIelement"

body force components

subscripts indicating node i and nonconforming mode j

Jacobian and determinant of Jacobian

stiffness matrix of structure and stiffness matrix of

an element

M , M , M )( Y xy

N, N

N , N , N x y xy

p~ 1

Q

R

r

S

s

T

to' t. 1

U, U. 1

IT

12

submatrices of Ke

condensed element matrix

moments in x and y directions and twisting moment

shape functions in curvilinear coordinates, conforming and nonconforming, respectively

forces in x and y directions and shear force

generalized load at node i in an element

interpolation functions for surface tractions

intensity of uniformly distributed load and intensity

of load at node i

load vector

boundary tractions

surface

corrective factor for shear

surface traction

uniform thickness of shell and thickness at node

nodal displacement vector and displacement at node i

amplitude of nonconforming displacement modes

u, v, w

I

U , V , W

v

cSw

X, X. 1

x, y, z

, I

X , Y , z

Ct, B

E, E

Ex,Ey'Yxy '

Yxz,Yyz

8

KX I 'K Y I 'K)( I Y

13

components of displacement in the x, y, z directions

components of displacement in the x , y , z directions

volume

J I

unit direction vectors in the x , y , z directions,

respectively

virtual work

global coordinates and coordinates at node i

global coordinates (components of X)

local coordinates

J

rotations about the x ,y axes

strain vector in global and local coordinate systems, respectively

components of E

curvilinear coordinates

direction cosine matrix = [v l ' v2' v3 J

curvature changes of middle surface of shell

Poissonis ratio

potential energy function

0, 0

cp. 1

14

stress vector in global and local coordinate system, respectively

=

15

Chapter 2

ISOPARAMETRIC ELEMENTS

2.1 General

One of the approaches to obtain the desired displacement field is

to represent it by Hermitian interpolation (or shape) functions [llJ as has

been discussed in the previous chapter. The similar approach using Lagrangian

interpolation functions may be used to begin the development of the isopara-

metric element, so designated because it uses the same shape functions for

representing both the global coordinates and the displacements. Zienkiewicz,

Irons and their co-workers at the University of Wales, Swansea [39-41,47,48J

are the most prolific contributors to the development and application of this

type of element.

The basic concept and derivation of this element has been well

publicized and one need only refer to Zienkiewicz's textbook [47J or the

references cited above for complete details of the basic approach. Therefore,

a detailed development of the basic element will not be presented here. How-

ever, the significant characteristics of the isoparametric element are sum-

marized below:

1 . The same shape functions are used for defining the geometry

and the unknown displacement variables.

X I N. 1

x. (2. 1 ) 1

U L N. 1

U. 1

(2.2)

16

where X = global coordinates

N. = shape function at node i (Appendix A) 1

x. = coordinates at node i 1

U = global displacements of any point in the element

U. = displacement at node i 1

2. Gausian quadrature is the only practical method to calculate

the element properties.

3. Geometrical conformity and displacement continuity are satisfied

after deformation.

4. The constant derivative condition is satisfied for all iso-

parametric elements. This means that rigid motions do not

cause strain.

5. In order to calculate the stiffness matrices~ transformations

are necessary to evaluate the global derivatives from curvi-

linear derivatives and to perform volume integration.

The isoparametric element is most advantageous when applied to three

dimensional structures, but it can be extended to plate and shell problems as

well. In this study the investigation is limited to plate and shell elements

derived from the three dimensional isoparametric element.

2.2 Isoparametric Shell Element

A three dimensional "Brick Type fl isoparametric element (Fig. 1)

may be used directly for plates and shells. If the thickness of the ele-

ment is reduced enough in comparison with the element side length it will

actually function as a plate or shell element. This type of element, however,

17

has two weaknesses or faults which restrict its practical application. These

deficiencies are:

1. The element is too stiff against flexure. This is the result

of IIparasitic shears ll which develop because of the constraints

placed on the transverse displacement mode. The phenomenon is

particularly severe for low order elements. Figures 2b and 2c

show the constrained mode and the true mode of the linear ele­

ment when subjected to pure bending.

2. Too large a number of variables have to be included as com­

pared with conventional shell elements. For instance, the

cubic isoparametric element (Fig. lc) has 32 nodal points which

translates into a total of 96 unknowns while most conventional

quadrilateral shell elements have only 20 unknowns. This mul­

tiplication of unknowns results from not taking into account

such things as normals remaining normal.

So far, no specific assumptions which a conventional shell element

utilizes have been made. Therefore, transverse shear deformation is taken into

account for this element. The retention of the shear effect is important

particularly when the element is used for thick shell problems where the trans­

verse shear effect is no longer negligible.

The correction of the deficiencies of the isoparametric element given

above is the subject of the following sections.

2.3 Ahmad-Irons Modification

The need to require an excessive number of unknowns, listed as

second undesirable feature of original isoparametric shell element may be

18

corrected in part by the application of the well known Kirchoff's hypothesis,

i.e., the normal to the mid-surface remains straight and normal after defor-

mation with the corresponding strain energy ignored. Thus, the unnecessary

nodes thru the thickness of the element can be eliminated (Figs. ld and l~).

With this linear expansion in the normal direction ~ there remains now a

total of six degrees of freedom for each set of top and bottom nodes (Fig. 4a).

Ignoring the elongation in the normal direction ~ in addition to

the above assumption, Ahmad et ale [40J replaced the six degrees of freedom

by three mid-surface displacements (u, v, and w) in the global directions

(x, y, and z) and two rotational displacements (a and S) around the two

1 ()r~ 11 \I rlo-f"i nor! :::l voe (v I ::> nrl \I I \ • ' .. n ......... 'J '-"'- I • "'-,"" UA'-..) \ A UIIU.J I. Thus the element has five basic degrees

of freedom at each node just as a conventional shell element would have. The

definitions of the curvilinear coordinate system (~, n and ~), the global I i

coordinate system ~,y, and z) and the localized coordinate system (x , y , i

and z ) are""given in Appendix B.

Now the Cartesian coordinates (x, y, and z) of any point in the

shell element is expressed thru its curvilinear coordinates (~, n and ~) and

the mid-surface global coordinates. Thus Eq. 2.1 of the original isopara­

metric element is changed [40J to the form

x X. 1

Y L N. y. + L N ~ \/3i (2.3) 1 1 i 2

z Z. 1

where the vector \/3" " 1 1.s given as

19

x. X. 1 1

V3i = y. y.

1 1

Z. z· 1 J top 1 bottom

For a shell with constant thickness, t , o

and

N. = shape functions in curvilinear coordinates 1

to = constant thickness of shell

v3i unit vector normal to the shell surface at node i

(2.3.1)

(2.3.2)

The corresponding displacement components (u, v and w) in the

directians of global axes (x, y and z) are expressed in the form

where

u u. 1

-:: ] 1

L N. L N. £ t o cP i (2.4) v = v.

I +

1 1 1 2

w W. 1 J

I I

a.,S· = rotations around the axes (x ,y ) in the directions of 1 . 1

vl and v2 respectively at node i

cP . 1

=

One may refer to Appendix B of this thesis for the reference axes I I

(x ,y and z ) and their direction vectors (vl ' v2 and v3) and Appendix A

for the definition of the shape functions. It is noted here that the shape

20

functions in Eqs. 2.3 and 2.4 are defined by only two coordinates (~,n) while

those of Eqs. 2.1 and 2.2 are defined by three (~, nand s).

This element substantially reduces the number of unknowns from the

original isoparametric element, but it still retains most of the character­

istics of the original element. Therefore, the first mentioned deficiency

of the original element (that it is much too stiff in bending) has been left

virtually untouched.

Some numerical results using linear, quadrilateral and cubic ele­

ments are given in Fig. 10 to demonstrate the behavior of this type of ele­

ment. The figure shows the central deflection of a clamped square plate

under a concentrated load placed at the center plotted against the mesh size

for the various elements. As expected, only cubic element shows fairly good

convergence to the real solution while quadratic element is rather slow in

converging and the linear element behaves so badly that, unless the sub­

division is reduced to uneconomical limits, the element is not applicable

to a plate or shell problem.

, Schemes for the improvement of this weakness are discussed in the

following two sections where a reduced integration technique and the addition

of nonconforming displacement modes are described.

2.4 Reduced Integration Technique

It has been noted in the previous section that the low order ele­

ment is far too stiff to be an economical element to use when the depth of

the member is small compared to its. plan dimensions. Figure 2 explains this

source of error present in the use of the element. The constraints in the

21

normal direction n require the strains to be as shown in Fig. 2b. Associated

with this configuration is the tendency to absorb too much of the deformation

energy by shear energy.

One of the ways to overcome this difficulty is the elimination of

the excessive shear energy by the means of a selective reduction of the

order of numerical integration. This technique was first introduced by

Doherty et al. [49J and later elaborated on by Pawsey [50J. This procedure

was also investigated independently by Zienkiewicz et al. [41J. Marked im-

provement was shown in the problems these groups investigated.

The.basic reasons why the simple reduction in integration order

produced such improvement are:

1. The reduction in the order of integration tends to reduce

the value of stiffness of an element below the value exact

integration produces. Since the element starts too stiff,

the softening has desirable effects.

2. The reduced integration can exclude the regions where the dis­

placement may be over constrained. For the linear element (Fig.

2a), for instance, a one point numerical integration at the center

of the element where the shear strain is zero results in zero

shear energy. Similar reductions occur with the quadrilateral

element when a 2 x 2 'integration is used.

Obtaining the stiffness matrix of an element involves evaluating

an integral of the general form

r r r T J J J B d B IJI d ~ d n d ~ (2.5)

where

22

/JI = determinant of Jacobian

D = material property matrix

B = matrix relating nodal variables and strains

The integration is most readily carried out numerically using Gaussian

quadrative with an integration order of 2 x 2, 3 x 3, or 4 x 4 mesh of Gauss

points for the linear, quadratic or cubic elements respectively [50J. These

integration points are necessary to evaluate the volume of an element ade­

quately.

As pointed out in the beginning of this section, the source of the

errors (Fig. 2) is excessive shear energy. Therefore, the reduced integra­

tion order needs to be applied only to the transverse shear component while

maintaining the usual complete integration on the other strain components.

The integration procedure, however, should be divided up so that each com­

ponent of strain energy is integrated over its own grid of integration points.

Although the selectively reduced integration improves the perfor­

mance of the element [41,50J, a uniform or all-round reduction of integration

order, for instance, 2 x 2 integration on all the strain components for a

quadratic element, produces even more rapid convergence and better ac-

curacy [41J. The reasons of such an improvement, however, are not readily

explained mathematically.

Two-fold benefits accrue from this technique. These are the im­

provement in accuracy but also coupled with a saving in the time required

to calculate the element stiffness.

2.5 Addi.tion of Nonconforming Displacement Modes

Another general approach to improve the basic accuracy of the

23

isoparametric element by eliminating the excessive shear energy is thru the

addition of nonconforming modes. Instead of disregarding the shear energy

during the integrating procedure, Wilson et al. [42J adopted a new approach,

that of eliminating the transverse displacement constraints which cause the

excessive shear strain in the element. This is achieved by adding extra dis-

placement modes. The additional displacement modes are of the same form as

the errors in the original displacement approximation.

A two-dimensional element under pure bending, for instance, is

shown in Fig. 2a. The exact displacement field for this type of loading

(Fig. 2c) is of the form

where

u

v 1 2 '2 (Xl (1 - E;, )

2 + (X2 (1 - Tl )

(Xl amplitude of the selected displacement shape

= a function of material properties, for w = 0, cx2

(2.6)

(2.7)

o

The linear isoparametric element is capable of representing the

displacement component u (Eq. 2.6) but not capable of including the trans­

verse displacement component v (Eq. 2.7). This constraint in the transverse

direction results in extraneous shear strains (Fig. 2b). What was lacking

in the representation of the displacement field by the isoparametric finite

element is the very source of the error. The addition of extra displacement

modes in the same form as Eq. 2.7,

v + 2 13 2 (l - Tl ) (2.8)

24

results in the elimination of the extraneous shear strain and the restoration

of the actual behavior of an element (Fig. 3). These additional modes gen-

erally destroy the compatibility of the isoparametric element.

Now the total displacement field of any element can be expressed

by adding the additional modes to the original isoparametric element displace-

ment field, Eq. 2.2. Thus,

where

u \ N. U. + L\ N. U. L 1 1 J J

N. additional nonconforming displacement modes J

U. = additional unknowns corresponding to the additional J

displacement modes

(2.9)

These additional unknowns can be taken as the displacements of a

selected set of internal modes of the element simply as a set of generalized

coordinates associated with the respective nonconforming modes. The addi-

tional unknowns are also determined by minimizing the strain energy in the

element. The resulting stiffness matrix, using the displacement field ex-

pressed by Eq. 2.9, has been enlarged over the usual isoparametric element

matrix due to the additional modes and the corresponding unknowns. This

enlarged matri~ however, can be partitioned then condensed back to the order

of the original matrix, i.e., without the additional modes, by means of

static condensation. This is possible since the additional unknowns occur

only in the equations applicable to the specific element into which the non-

conforming modes have been applied.

25

The examples given by Wilson et al. [42J demonstrate marked im­

provement over the low order element. Convergence to the true solution is

obtained. Their work~ however, was limited to the cases of plane strain and

thick shell problems. The necessity of developing a new general thin shell

element still exists.

The following chapter deals with the development of elements based

on the approaches discussed in this chapter.

26

Chapter 3

ELEMENT FORMULATION

3.1 Variational Principle

As seen in Section 1.2, the variational principle has been fre-

quently used as the basis of element formulation [2J. For a typical displace-

ment model, without any boundary constraints, the total potential energy

functional TIp may be expressed, utilizing tensor notations, as

where

- f T.U. d5 50 1 1

TIe = potential energy functional for element e p

s. . = strain tensor components lJ

Eijkl = elastic constants

v

F. 1

U. 1

5

T. 1

=

=

=

=

=

volume

body force component

displacement

surface

surface traction

(3. 1 )

50 -" portion of 5 over which the surface tractions are pre-

scribed

27

The displacement U is defined by suitable functions expressed in

terms of variables at nodes, i.e.,

U = [N] U. 1

E = [B] u. 1

where subscript i indicates node i, not tensor notation, and

[N] interpolation functions

[B] matrix relating strains to nodal displacements

Substituting Eqs. 3.2 and 3.3 into Eq. 3.1, one gets

It p r

= J v

(-21 U~ BT EBU. - U~ NT F) dV -1 1 1 J

r U ~ NT T dS So 1

(3.2)

(3.3)

(3.4)

where the subscripts for tensor notation have been dropped for reasons of

.simplicity. This equation can be written in a more compact form as

where

Iv BT E B dV

J NT F dV +

v NT T dS

(3.5)

(3.5.1 )

(3.5.2)

e e K and R are referred to as element "stiffness matrix" and "load vector,'1

respectively.

Taking variations with respect to the nodal variables U., then 1

Eq. 3.5 yields

o (3.6)

28

Assembling the element stiffness matrix and load vector into whole structure,

KU R = 0 . (3.7)

Thus a system of linear algebraic equations, which is known as the

IIload-displacement equation,1I is derived.

3.2 The Basic Geometry and Displacement Field

The isoparametric shell element and some improvements and modifi-

cations to it have been described in the preceding chapter. Some of the con-

cepts in the element development as described are directly utilized in this

study. The Ahmad-Irons definitions for the shell geometry and displacement

field, Eq. 2.3 and Eq. 2.4, respectively, are taken as the starting point of

this study, The equations expressing these definitions are rewritten here

as

and

r : L

Z

u

v

w

X. 1

= L N. y. 1 1

Z. 1

u. 1

= L N. v· 1 1

w· 1

(3.8)

t. [:',.' 1 + I Ni -+ L, <Pi JJ J (3.9)

It is noted again that if the shape functions N. are such that no 1

volume discontinuity or gap or overlay is produced along interelement bound-

aries by unique coordinates at the nodes, the displacement compatibility is

also maintained at the interelement boundaries.

29

The expression defining the mapping of the element as indicated by

Eq. 3.8 is used throughout this study without any modifications to represent

the global coordinates of any point of the shell. However, the equations ex-

pressing the displacement such as Eq. 3.9 are used only as a starting point

in this study and subsequent modifications are made to these equations.

3.3 Nonconforming Displacement Modes

As discussed in Section 2.5, one of the techniques to get rid of the

excessive stiffness of the isoparametric shell element is thru the addition

of nonconforming displacement modes [42J. Thus, by sacrificing the inter-

element displacement compatibility present in the original isoparametric

element, it becomes possible to eliminate or at least minimize the built-in

constraints that the element has against the transverse displacements. It

was shown in Section 1.2 that for displacement models strict interelement

boundary displacement compatibility is not essential in order to assure

convergence. This fact is used in this study as the basis for relaxing the

strict compatibility requirement.

Thus, in addition to the basic shape functions defined in Appendix

A, the following nonconforming displacement modes are introduced. For a

linear element, the nonconforming shapes to be appended are

Nl = ~2

N2 2 = n

N3 = n (1 ~2) (3.10)

N4 = E,; ( 1 n2)

N5 ( 1 - E,;2) (1 ? ,

= n-)

30

while for a quadratic element~ the shapes are expressed by

Nl = ~ (1 ~2)

N2 nO 2 = n )

N3 ~n(l ~2) (3.11)

N4 = n~(l n2

)

N5 ( 1 n2) (1 _ ~2)

As seen in Section 2.3 (Fig. 10), since the cubic element behaves in

an adequate manner and there is already a sizeable number of degrees of freedom

associated with that element, no additional displacement modes are or should

be added to improve its behavior. The cubic element is used primarily to

evaluate the accuracy of the new nonconforming elements by comparison.

The additional modes in Eqs. 3.10 and 3.11 are so selected to have

zero values at each node and to eliminate the undesirable constraints present

in the original isoparametric shapes. For instance, the first two modes in

Eqs. 3.10 are designed to eliminate the transverse displacement constraints

in the same way as was discussed in Section 2.5. The 3rd and 4th modes in

Eqs. 3.10 contribute to the softening of the twisting constraints while the

5th mode adds the bubble shape that provides for displacements totally within

the region.

The additional modes for the quadratic element~ as listed in (Eq.

3.11), are selected along the same lines as for the linear element. These

modes are illustrated in FigS. 5 and 6 for linear and quadratic elements,

respecti ve ly.

Now the general form of the displacement field with nonconforming

31

modes included can be formed by adding the extra modes of Eq. 3.10 or Eq.

3. 11 to Eq. 3.9. Thus,

'\ N. L 1

u. 1

v· 1

W. 1

+

u. J

v. J

+

w. J

(3.12)

Equation 3.12 is the general case where all five additional modes

(Eq. 3.10 or 3.11) are applied to all five variab1es (u,v,w, aand G) at each

node. It may be unnecessary, even excessive, to carryall nonconforming

modes for all displacement components. Slower convergence may result or the

highly nonconforming element may prove uneconomical. The optimum combination

of additional modes for each nodal variable is investigated in Section 3.7.

3.4 Strain-Displacement Relationships

Once the displacement field is defined, the strain-displacement relation-

ship is established by combining the proper derivatives of the displacement com-

ponents. If the basic shell assumptions are adopted, it is most convenient to

erect a local coordinate system (Xl, yl, Zl) as defined in Appendix B and to take

the displacement derivatives with respect to this coordinate system.

The components of strains in the local coordinate system are

32

I

au E: I

D

X ax I

av E: I

I Y ay I B

Y I I ~+~ { 11 x Y I I

(3.13) E: J = = ay ax I I

Yx'z I aw + ~ I I

ax az I I

Y I I ~+~ y.. Z I I

ay az

where E: I is neglected in order to be consistent with the assumptions made . z in Section 2.3.

Before obtaining the strain vector (Eq. 3.13), it is convenient to

establish first a matrix consisting of the local derivatives of all displace-

ment components (Eq. 3. 14) . The terms needed for the strain vector can then

be selected and assembled.

r~ I I

1

av aw I I

ax ax ax I I !

au av aw (3.14) I I I

ay ay ay I I I

au av aw J I I

az az az

Since the displacements are defined in a global coordinate system

(Eq. 3.9) and the shape functions, N. and N. in Eq. 3.8 and ·Eq. 3.12, are 1 J

expressed in terms of curvilinear coordinates (~,n,~), two elaborate trans-

formations are necessary in order to obtain the strains in a local coordinate

33

system. The operations of these transformations are given in Appendix C.

The resulting expressions for strain components in the local coordinate sys-

tem with the global displacements at the nodes are

u.

cis\[:;j u.

[ ~~ J 1 t. J t.

{c:} B.8 T 1 B.8T J - T v. + 2 + v. + -2 C.8 cpo

1 1 J J J J

w. W. 1 J

(3.15)

where B., C., B. and C. are defined in Appendix C. 1 1 J J

For simplicity the strains in Eq. 3.15 are expressed with two sep-

arate parts, i.e., conforming (ordinary isoparametric element) and noncon-

forming parts; thus

{c} [8, -B ] l ~ 1 (3.16)

3.5 Stress-Strain Relationships

The stress components corresponding to the strain components are

given by

a i -1

X

a I y

I

{a } = a I I

x y (3.17)

o "j x z

a I I

Y z

I . I

where the stress vector {a } and material property matrix [0 ] are expressed

34

again in the local coordinate system (Appendix B). It is also noted that

is neglected to be consistent with shell assumptions.

o I

Z

For simplicity, a homogeneous linearly elastic material whose pro-

perties are the same in all the directions is considered. For this material I

the matrix [0 J is of the form

where

[O'J E 2

- ]J

l sym~ E Young's modulus

]J = Poisson's ratio

- ]J

2 (3.18)

A numerical corrective factor s is needed since the actual shear

distribution across the thickness is quadratic whereas a constant (or aver-

age) shear distribution is assumed as a consequence of the shell assumptions.

The ratio of actual shear strain energy to the assumed shear strain energy is

approximately 6/5 for rectangular beams [51J and this corrective factor is

adopted in this study.

3.6 Element Stiffness Matrix

I I

Once the material property matrix [0 J and the strain vector {€ }

are obtained, the element stiffness matrix is obtained by the direct application

35

of variational principles as shown in Section 3.1. The thus obtained stiff-

ness matrix is given in Eq. 3.5.1, or

(3.19)

where the superscript e denotes the individual element and the volume dV has

to be transformed for the integration process as

dV = dx dy dz IJI dt;: dn de: (3.19.1)

where IJI = determinant of the Jacobian matrix.

Equation 3.19 is now of the form

(3.20)

The explicit integration of this equation is so complicated that

numerical integration is the only practical way to achieve the integration.

An outline of the Gaussian numerical integration method is given in Appendix

D. More details can also be found in References 47 and 50.

With the displacement vector U and the [S] matrix subdivided into

conforming and nonconforming parts, then Eq. 3.7 may be rewritten in the form

::: 1 [~l = [: 1 (3.21 )

where

r ST E S dV Jv

(3.21.1)

f ST E B dV (3.21.2) v

36

K22 ~ I v B T E B dV (3.21.3)

The submatrix Kll is the ordinary isoparametric element stiffness

matrix and Kl2 and K22 have been formed from the nonconforming displacement

modes. Thus the element stiffness matrix has been enlarged. The expressions

for typical components of the element stiffness matrix are given in Appendix

E.

Separating Eq. 3.21 into two matrix equations

Kl1U + Kl2U = R (3.22)

T -K12~ + K22U = 0 (3.23)

The nonconforming modes are condensed out first. Solving Eq. 3.23 for U,

the amplitude of the additional modes are found to be,

U = (3.24)

On substituting IT into Eq. 3.22, the variables for the nonconforming displace-

ment modes are eliminated. The final set of equations to be used is

i

K U R (3.25)

where I

K = (3.25.1)

I

Thus, the newly formed element stiffness matrix K has been con-

densed back to the same' order as an ordinary isoparametric element stiffness

matrix Kll"

37

It should be noted, however, that the advantages which the additional

nonconforming modes provide may not be worth the cost if the operations in-

vo1ved in forming matrices K12 and K22 and in subsequently condensing them I

back out to form matrix K (Eq. 3.25.1) are too expensive. Since the cost of

calculations depends on the size of the additional matrices, K12 and K22 , it

is desirable to reduce the order of such matrices as much as possible without

hurting the accuracy of the nonconforming element.

This can be achieved in part by applying the nonconforming modes

selectively to the various variables in much the same fashion as selective

reduction of integration points (Section 2.4) has been used.

3.7 Selection of Nonconforming Modes

Thus far it has been assumed that all five additional modes in Eq.

3.10 or 3.11 are applied to all five variables at each node, Bakhrebah [52J.

Examining the behavior of the isoparametric elements (Fig. 2), however, it

can be observed that not all the displacement components are constrained to

the same degree. While the transverse displacement in Fig. 2 is severely

constrained, the longitudinal displacement may be adequately represented by

the linear variation present in the isoparametric element.

This indicates the strong desirability of using a selective addition

of nonconforming modes. The reduction of additional modes applied to the

conforming modes is particularly important because the numerical integration

and matrix condensation procedures in forming the final stiffness matrix are

expensive and maybe more so than is necessary.

For a linear element, inclusion of the fifth mode of Eq. 3.10 re-

sults in a singular matrix for K22 . Therefore, that mode is deleted from

38

further consideration. The next question raised is the validity of applying

the nonconforming modes to the rotations. As seen in Section 2.3 the rota­

tions at the nodes (a and S) are approximated as the average difference, in

inplane displacements (u and v) across the thickness. Thus, the rotations

at node i are expressed by

(v . v. ) ltop lbottom

a :::::

t. 1 (3.26)

(u. u. ) S ::::: 'top 1 bottom

t. 1

According to the approximations in Eqs. 3.26, the rotations are

established solely from the inplane displacements. Therefore, if the inplane

displacements are satisfactory with the original element, no additional ro-

tational nonconforming modes are necessary or even desirable for the new

element.

It has been seen that the parity of orders of polynomial represen-

tations for the displacement components u, v and w is not essential for a dis-

placement model [34J. A linear expansion for displacement components u and v

and cubic expansion for w has been used with a great deal of success [32,33J.

This should also hold true for nonconforming elements. The linear represen-

tation of the u and v displacement component is exact for the case of constant

moment and quadratic representation should be sufficient for linearly varying

moment. Therefore, the addition of the first two nonconforming modes of Eq. )

3.10 to the displacement components u and v should be sufficient approximation

for a linear element. For a quadratic element which already has second order

39

terms in representing u and v displacements, the addition of the nonconforming

modes to u and v displacement components is less important than in the case of

a linear element.

Finally, the vertical displacement component w can be approximated

adequately by a cubic representation. For both the linear and the quadratic

elements, the inclusion of the 3rd and 4th modes from Eq. 3.10 or Eq. 3.11

(twisting modes) are essential to eliminate the twisting constraints. The

decision as to whether the 5th mode of Eq. 3.11 is essential to improve the

accuracy of the quadratic element is difficult. Further discussion of this

matter is delayed to Chapter 5 where the numerical results are available to

form the basis of a judgement.

From among the possible nonconforming modes those considered to be

most beneficial have been predicted to be NC4-4.3R for the linear element and

NC8-4.3R or NC8-5R for the quadratic element (see Table 1). For both the

linear and the quadratic cases, the reduction of size of matrices due to the

addition of only selected nonconforming modes is significant. For instance,

the order of matrix K22 in Eq. 3.21 is 25 x 25 for both linear and quadratic

elements using all nonconforming modes and before the reduction. This matrix

K22 is reduced to order 8 x 8 after selected reduction.

3.8 Forced Interelement Boundary Compatibility

Due to the introduction of nonconforming displacement modes (Sec­

tion 3.3), the element used in this study starts with a violation of compati­

bility along the interelement boundaries. This may be in the form of gaps,

overlaps and/or kinks along these boundaries. Initially, it was considered

40

desirable to re-establish compatibility along these boundaries. One of the

ways to re-establish this interelement boundary compatibility is by the means

of self-equilibrating boundary tractions which push, pull and/or rotate the

boundary back to the compatible position.

For this case, the element described in this study becomes equiva-

lent to that known as.a hybrid displacement model. For a hybrid displacement

model, the equations equivalent to Eq. 3.7 are given as

KU R + T C r = o

c U = '0

or in matrix form

where

cT = I N Q dS as

Q interpolation function for boundary tractions

r = nodal tractions

as = portion of boundary where the tractions are applied.

(3.27)

(3.28)

The matrix C is known as a constraint matrix. It is simply the statements

of compatibility along the interelement boundaries.

The second equation of Eqs. 3.27, which is a system of homogeneous

linear equations, generally does not have unique solutions except zeros for

all unknowns U. Of course, if the matrix C has a rank (r) less than the

number (n) of unknowns U, some r of the unknowns can be solved in terms of

41

the remaining n - r unknowns. In this case, there are infinite number of

solutions. Substituting these solutions into the first equation of Eq. 3.27~

the final solutions can be obtained. The procedure, however, is not as simple

as setting up the equations themselves.

Another alternative approach is to eliminate U before r in the

equation solving process. Solving the first equation of Eqs. 3.27 for U,

(3.29)

and substituting U into the second equation of Eq. 3.27

= o (3.30)

Solving the Eq. 3.30 for r and substituting it into Eq. 3.29, one gets nodal

displacement U and nodal traction r. The solution is costly, however, mainly

because of the necessity to invert the stiffness matrix K and in the process

loose the banded character of the enlarged matrix to be solved. This could

be avoided if the equations were properly ordered in the machine with the

constraint equations first and the variables in proper order.

In this study it was originally intended to re-establish complete

interelement compatibility by the means of boundary tractions thru the process des­

cribed above. This operation being applied after having added the nonconforming

modes which destroyed the compatibility to begin with. The performance of the

nonconforming modes, even 10111 order element, however, was so satisfactory that

it has been found unnecessary to apply the constraint equations.

For the above two reasons, i.e., cost of the equation solving pro-

cess and the already adequate nature of the solution, the idea of re-establishing

compatibility of the interelement boundaries has been abandoned in this study.

42

Chapter 4

ASSEMBLY AND SOLUTION OF THE LOAD-DISPLACEMENT EQUATIONS

4.1 General

By the application of one of the variational principles (Mi~imum

Potential Energy Principle in this study) the general form of load-deflection

equation was obtained in Section 3.1.

KU R = 0 (4. 1 )

In this systems of lfnear simultaneous algebraic equations, the

stiffness matrix of the whole structure K and the load vector R are known.

Therefore Eq. 4.1 is solved for unknown displacement vector U.

Once the displacement U is obtained, the corresponding strains c

and stress 0 and/or forces can be obtained by the relations of Eq. 3.16 and

Eq. 3.17, respectively.

4.2 Assemblage of Stiffness t-.1atrixand Boundary Conditions

The stiffness matrix for the total structure K is formed by summing

the contributions of all the individual element stiffness matrices. This

initial stiffness matrix would be singular. The final assembled stiffness

matrix K must reflect the boundary conditions. Then Eq. 4.1 becomes solvable.

Boundary conditions are incorporated by deleting or modifying the

appropriate rows and columns from total stiffness matrix K corresponding to

the dis~lacementS·constrained or specified at appropriate nodes. The deletion

can be performed effectively during the stiffness matrix assembling process.

43

One effective method of assembling the stiffness matrix is the code number

system discussed by Tezcan [64J.

4.3 Load Vector

In the finite element formulations, the applied load must be trans-

formed to the nodal loads. Though the generalized load is the theoretically

correct and more acceptable load to use the statically equivalent load can

be used satisfactorily [34J.

The generalized load is defined as the equivalent nodal loads which

produces the same amount of work done as the actual loading would when moved

through the corresponding virtual displacement. For distributed loads, the

virtual work is expressed by

where

p~ 6u 1 f q. N. dA 6u

All

p~ = generalized load at node 1

6u = virtual displacement

q. intensity of load at nodes 1

N. shape function 1

in an element

cancelling the virtual displacement 6u in Eq. 4.2.

p~ 1

r

J q. N. dA

All

For the uniformly distributed load q , o

p~ 1

r qu- J N. dA A 1

(4.2)

(4.3)

(4.4)

44

Again the integration is carried out by the numerical integration method just

as with the element stiffness matrix.

The assemblage of load vector R can be performed in the same way

as the stifness matrix assemblage. The loads corresponding to the constrained

displacements should be omitted during assemblage.

4.4 Solutions of Load-Deflection Equations

Solving the simultaneous algebraic equations Eq. 4.1 following the

formation of structural stiffness matrix, is the central part of finite ele­

ment method. Therefore, in order to achieve the economy of the method, it is

essential to take maximum advantage of well-known special characteristics of

the stiffness matrix, i.e., symmetric, banded and positive definite.

Gauss elimination procedure takes fewer operations than other

method using rational operations for a dense matrix of coefficients [54J.

One of the most efficient modifications based on Gauss elimination is one

known as the "frontal solution technique lf [55J.

Instead of using a band algorithm, the frontal method uses the

specific IIfrontl! concept. As the front advances, the element stiffness co­

efficients are assembled and the variables which appear for the last time

at that front are separated and eliminated immediately. This elimination

process is the same way as the Gauss elimination. The size of the Iffront,"

however, is smaller or, at worst, same as the band width of the equation.

The frontal solution is one of the more effective methods for

element~ with nodal points at more than just the corners, or three dimensional

elements whose band width becomes very large. Branching structures is another

45

class of problem where the frontal method posses advantages. -For the element

with nodes at corners only this approach may not be as effective as for the

elements mentioned above due to the large input-output requirements of the

method. The frontal solution is adopted as the equation solver in this study.

4.5 Strains

The strain-displacement relationships have been established in

Section 3.4 (Eqs. 3.15 and 3.16).

(4.5)

Thus the local strains are calculated with the global displacements.

If only the values at the nodes are desired, then Eq. 4.5 can be

simplified by omitting the amplitudes of the nonconforming nodes IT and the

corresponding B. Since the nonconforming displacement modes have little

direct effect on strain values computed at the nodes, as defined in Section

3.3, U'and B need not be calculated. It achieves substantial savings in

calculation effort. Thus the nodal strains are now obtained by

I

{s} = [B] {U} (4.6)

It should be noted, however, that if the values are desired at

points other than the nodes, such as the Gaussian integration points, then

Eq. 4.5 with the nonconforming contribution rather than Eq. 4.6 is used in

order to obtain the best values at those points.

4.6 Stresses and Forces

Stresses in local coordinate systems are obtained by Eq. 3.17, i.e.,

46

I I •

{o } [0 ] {E } (4.7)

For the thin shell problems, it is convenient to obtain the forces in the

shell coordinate system. The axial forces per unit shell width are obtained

by the relations

And the moments are obtained by

where

M I = H(K I + 1-1K I) X X Y

M I = H(K I + 1-1K I) Y Y x

M I i = H( 1 ;1-1)(2K I.) X Y X Y

H = 2 12(1 - 1-1 )

K I, etc. x =

(4.8) ,

(4.9)

The approximations N ' I = N I I and M 8 I =- MIl are applied in x y y x x y y x

accorda~ce with thin shell assumptions.

47

Chapter 5

NUMERICAL RESULTS

5. 1 Genera 1

The series of numerical examples presented in this chapter are

1) to provide a numerical background for selecting the most beneficial COI11-

binations of nonconforming modes (Section 3.7), 2) to demonstrate the improve­

ment of accuracy obtainable by the addition of these nonconforming displacement

modes and 3) to demonstrate the applicability of such a procedure to practical

problems.

In order to discuss the various combinations of nonconforming modes,

a special notational system was established. The designations for some of the

elements under consideration are tabulated in Table 1 using the notational

system described below. Taking NC4-4.3R, for instance, the NC4- indicates

the element is a 4- noded element (linear element) with ~on~onforming dis­

placement modes. The figures _~~_ indicate that ± nonconforming modes (Eq.

3.10) are applied to 1 variables (u, v, w). Finally R indicates that a Re­

duced number of nonconforming modes, in this case 2 out of possible 4 modes,

are actually applied to the inplane displacement components while all 4 non­

conforming modes are applied to the normal deflection w (Table 1). For another

example, C4 indicates that the element is a fonforming ~-node element, i.e.,

original isoparametric element with no additional nonconforming modes.

A number of different types of problems were studied to assess the

selective nonconforming approach to the element development. In order to

evaluate inplane behavior and simple bending behavior of the various nonconforming

48

elements, a deep beam was investigated for various load combinations. Next

clamped rectangular plates were investigated for a concentrated load applied

to the center. These tests evaluate the plate bending behavior of noncon-

forming elements. All major elements listed in Table 1 have been used in

the above listed problem series, the results are compared and the most bene-

ficial elements chosen. Only those elements passing this preliminary screen-

ing were used in subsequent investigations.

A cylindrical shell and an elliptic paraboloid were investigated

to evaluate the basic behavior of nonconforming elements for single and double

curvature shells, respectively. A hyperbolic paraboloid, a spherical shell

and a conoidal shell were also investigated in order to demonstrate the versa-

tility of newly developed nonconforming elements.

5.2 Deep Beams

5.2.1 Cantilevered Beam

A deep cantilevered beam whose dimensions and material properties I

are shown in Table 2 has been examined for two different types of loadings.

The results are tabulated in Table 2 and compared with the "exact" solution

by Popov [51J in which the shear effect has been taken into account.

It is obvious that it is unfair to compare the linear (4-node),

quadratic (8-node) and cubic (12-node) elements in the terms of number of

elements. Such a comparison is grossly biased to the higher order elements.

An alternative is to use the number of unknowns involved. This comparison

is also given in Table 2. An argument can also be made for the basis of com-

parison to be total computer time, programming effort, etc.

49

Due to the shape of the element and type of loading, any element

including N5 (Eq. 3.11) was not considered here so evaluation of them was

delayed until the next section where the bending behavior of plates was in­

vestigated. The mesh sizes used are indicated in parenthesis in Table 2.

For this first loading, the structure is subjected to constant

bending. Except element C4, all the elements investigated gave very good

solutions, as expected. Element C12 gave a little higher values for both de­

flection and stress but these differences are negligible.

For the linear element, the improvement obtained by adding two non­

conforming modes is the most impressive. (Compare the results from NC4-2.3,

NC4-4.3R and C4 in Table 2.) Addition of another two nodes to inplane dis­

placement components (NC4-4.3) had no noticeable effect for the type of ap­

plied loading. Because the additional nonconforming modes involve additional

computation they actually have a harmful or degrading effect.

Since the eight-node element (C8) has a quadratic representation of

displacement in its development, it produces good solutions. The addition

of nonconforming modes (NC8-2.3, NC8-4.3R and NC8-4.3) or utilizing higher

order element (C12) produced the same accuracy as NC4-4.3R or C8. Going to

this additional expense then produced no net return.

Loading Case (B)

The structure is now under a linearly varying moment. Only the

element designated as C12 gave accurate results while other elements produced

50

slightly different results. This is mainly because only the element C12 is

capable of representing the parabolic shear distribution across the depth of

the beam.

The linear elements (NC4-2.3~ NC4-4.3R, NC4-4.3) lack the capability

of expressing the shear distribution thru the depth and approximate the verti­

cal displacement by a quadratic variation while the real variation is cubic.

However, this does not mean the other elements are to be discarded because

the improvement achieved by NC4-4.3R (NC4-2.3 is actually the same element

under inplane loading) can be seen to be almost as good as C12 element. Con­

sidering the fewer unknowns involved, the result obtained by NC4-4.3R is very

satisfactory.

It has been seen that both NC4-4.3R and NC8-4.3R produced the same

accuracy as NC4-4.3 and NC8-4.3, respectively. In other words, for inplane

displacements, the addition of two nonconforming modes should be sufficient

and the addition of another two nonconforming modes is either unnecessary or

undesirable.

5.2.2 Summary of Beam Investigations

The major findings form the limited beam investigations are sum­

marized as follows:

1. The addition of nonconforming displacement modes to the original

isoparametric elements gives generally markedly improved results,

i.e.~ approximately th~ same accuracy as that from a one degree

higher order element but at less expense. For instance, just

for inplane, the element C8 has 16 degrees of freedom while

51

NC4-4.3R has 12 degrees of freedom which is then condensed to

8 while NC4-4.3 has 16 degrees of freedom which is condensed to

8. The results from NC4-4.3R are close to those achieved by C8.

With these savings the difference of accuracy was 1 percent or so.

2 .. For both linear and quadratic elements, the addition of two non­

conforming modes (Nl , N2) is sufficient to improve the inplane

behavior of elements. Addition of another two nonconforming

modes (N 3, N4) is either unnecessary or undesirable. Therefore

the elements NC4-4.3 and NC8-4.3 are deleted from further con-

siderations.

3. The improvement of the linear element has been found to be most

significant. When smaller grid size, equivalent to the distance

between nodes in higher order elements, is used the element

NC4-4.3R gives as accurate a result as the quadratic or cubic

element.

5.3 Plates

In the preceding section, the inplane behavior of nonconforming modes

was examined. In order to examine plate bending the addition of nonconforming

modes to deflection wand rotations a and S was investigated.

5.3.1 Square Plate

A clamped edge square plate under a concentrated load applied at

the center was investigated. The geometry and material properties are shown

in Fig. 9. Since the exact solution to this problem is available [56J, a

S2

number of authors have selected this case to evaluate the basic accuracy of

their element.

The central vertical deflection as computed by the linear and the

quadratic elements along with various combinations of nonconforming displace­

ment modes is plotted versus the mesh sizes in Figs. 10 to 11.

For the linear element, the inclusion of the NS mode (Eq. 3.10) re­

sults in a singular matrix of K22 (Eq. 3.21.3). Therefore, the element which

includes this nonconforming mode, NC4-S.S, is excluded from further consider­

ation. The elements NC4-4.S and NC4-2.S, in which the nonconforming modes are'

applied to rotations, not only do not converge to the right solutions, but

also the magnitude of error is far too large (Fig. lla). As discussed earlier

in Section 3.7, rotations are established solely from the inplane displacements

and are not directly related to out-of-plane displacements. If the inplane

displacements are satisfactory, rotations should also be satisfactory (Fig.

2). Therefore, the addition of rotational nonconforming modes may result in

a more flexible structure than actual one.

The results obtained by the elements NC4-4.3R and NC4-2.3 are not

very good when a coarse mesh size is used. The ratio of convergence, however,

is very fast in both cases and very accurate results are obtained when the

grid has been refined to reasonable sizes. The element NC4-4.3R produced

superior results to the element NC4-2.3 because the latter does not include

the nonconforming twisting mode.

For the quadratic elements, the element which includes nonconforming

modes to the rotational degrees-of-freedom (NC8-S.S) behaves again much worse

than the elements which do not include the rotational nonconforming modes

{NC8-4.3R and NC8-S.3R}. Unlike the linear element, however, the magnitude

53

of the difference between the two, with and without, results is not nearly

as large as in the linear element case. The results indicate that the ele-

ment containing these nonconforming modes applied to the rotations (NC8-5.5)

has some tendency toward convergence to the real solution.

The results obtained by elements NC8-5.3R and NC8-4.3R are nearly

identical (Fig. llb). The former includes N5 mode while the latter does not.

The mode N5 has virtually no effect for plate bending problems.

The major findings of this section are

1. The application of nonconforming modes to the rotations may

results in either erroneous convergence (linear element) or at

least markedly slower convergence (quadratic element);

2. The fifth mode N5 is not appliable to low order elements due

to the singular nature of the matrix (linear element) or else

it has little effect and can be neglected for the higher order

elements (quadratic element);

3. With both linear and quadratic elements, the inclusion of twisting

modes (N3

and N4 for w) has significant effects and should not

be omitted.

As a conclusion the most rnmhin;:)+innc: \""'VIlIU I IIU v I VlloJ

of nonconforming modes are NC4-4.3R and NC8-4.3R for linear and quadratic

elements, respectively. Thus from here on only elements NC4-4.3R, NC8-4.3R

and C12 are considered. Although based on quite limited data, since the ele-

ment choice represents the intuitively more reasonable elements, and the cost

of continuing the examine the other elements was too high, the decision to re-

strict the study to just these elements is not premature. The designation of

these elements can now be simplified to NC4 and NC8, respectively.

54

The results of these elements compare favorably with those obtained

by other finite elements as shown in Fig. 12. It may be noted that results

from all the three elements (NC4, NC8 and C12) appear to converge to a value

slightly higher than the real solution. It is due to the fact that the use

of a minimum order of numerical integration in claculating the stiffness matrix

tends to under-estimate the stiffness of element (Appendix D). Since the dif­

ference is negligible and the use of reduced integration represents a signifi­

cant saving in time, its use is continued.

5.3.2 Circular Plate

The next test involves a circular clamped plate under a uniform

load. The main purpose of investigating this problem is to test the appli­

cability of nonconforming modes when used in conjunction with a nonrectangular

element arrangement.

As seen in Fig. 13, the solutions indicate good convergence char­

acteristics. Thus it is concluded that the nonconforming elements can be

used for both a rectangular element arrangement and for a nonrectangular ele­

ment arrangement. Marked departures from rectangularity should however be

avoided if at all possible.

5.4 Cylindrical Shell

A circular cylindrical shell supported by a rigid diaphgram and with

no longitudinal edge beam has become a standard test problem frequently inves­

tigated by a number of authors [32,40,57,58J. Thus the behavior of this type

of shell is well established. The purpose of investigating this shell in this

55

study is to evaluate the accuracy achievable with the newly developed ele-

ments when seeking solutions for singly curved shells.

The dimensions and material properties of the shell considered are

shown in Fig. l4a. Two loading cases as shown in Fig. l4b are considered, i.e.,

1) pressure load (47 psf on shell surface) plus live load (25 psf) and 2) dead

weight (150 pCf) plus live load (25 psf). The elements NC4 (10x8) and NC8

(5.4) are used with the mesh sizes indicated in the parentheses. Each was used

for both loading cases. The costs of computer usage by these two different

elements were approximately same.

The r,esults obtained are presented in Fig. 15 (displacements) and

Fig. 16 (stress resultants). These results are compared with a solution com­

puted by numerical integration of the Donnell-Jenkin1s equation [57J. Although

some differences are distinguishable in the results for the two loading cases, !

particularly in wand M I,for all intensive purposes the two cases yield the x

same result. Thus the approximation of the gravity load by a normal-to-surface

load which has been used previously by many investigators can be justified.

The agreement between results from Ref. 57 and those from NC4 and

NCB for loading case II is excellent with the exception of the hoop stress

N I and hoop moment M I. These stress resultants are rather sensitive for this x x

geometry and boundary conditions. The magnitude of the disagreement is more-

over relatively small and thus thought to be of small consequence.

The results obtained by NC4 and NC8 are generally in close agree-

ment for each loading case. However, the stress resultants obtained by NC8

at mid-side nodes, particularly N I in Fig. l6b and N I I in Fig. 16e are not x x y

so good as those at corner nodes. This results in a somewhat zigzaged nature

of plotted curve.

56

Another interesting~ yet useful, comparison to make is with the re-

sults obtained by Ahmad et al. [40J. They used 24 conforming cubic elements

(designated as C12 in this study, 60 OOF) and 24 conforming parabolic elements

(designated as C8 in this study, 40 OOF) in solving the same problem. It is

noted that the comparison with Ref. 40 is somewhat approximate rather than

exact because of a slightly different geometry (thickness to span ratio) and

loading used in Ref. 40. Whenever the difference between the exact solutions

cited in this study and in Ref. 40 is distinguisable in figures~ this differ-

ence in exact solutions is also shown.

The improvement in accuracy obtained by NC8 over C8 using a smaller

number of elements, 20 elements vs. 24, is very significant. The results ob-

tained by NC8 are as good as those obtained by C12. Considering the bad be-

havior of C4 in Fig. 10, the improvement achieved by NC4 is very impressive.

Thus the linear element which has been ignored by previous investigators has

been shown to be as useful as any other element.

5.5 Elliptic Paraboloid

A simp 1 y supported squa re ell i pti c pa raboloid whose geometry and ma-

terial properties are shown in Fig. 17 is investigated. Due to the square

plan form, the equation of the mid-surface of this elliptic paraboloid is

simplified to be

z 1 2 2 = - (x + y ) 2R

The shell is loaded by a uniformly distributed load.

(5. 1 )

The same shell was investigated previously by several authors [59,60J

57

using somewhat different approaches. This shell is analyzed in this study

in order to evaluate the basic accuracy of NC4 and NC8 elements for doubly

curved shells. Since the behavior of this type of shell, i.e., a shell with

positive Gaussian curvature, is predominately membrane action with bending

moments being mostly of a secondary stress nature and only significant near

boundaries, smaller mesh sizes are needed particularly near the edges of shell

in order to pick up the sharp stress gradient.

The results obtained by using NC4 and NC8 elements are shown in

Fig. 18 (displacements) and Fig. 19 (stress resultants). Reasonably good

agreement with the results from Gustafson and Schnobrich [60J and Chu and

Schnobrich [18J has been obtained with the exception of the shear N I I at x y

corner support. Since the plotted values of NC4 and NC8 are those at the

Gauss integration points, somewhat distant from the corner, these values may

not represent corner values adequately particularly where such a sharp stress

gradients exists.

The difference between the results obtained by using NC4 (8x8) or

by using NC8 (4x4) is again minimal. The mesh sizes are indicated in the

parenthses. Both NC4 and NC8 with above mesh sizes not only produced very

close accuracy but also they used nearly the same computer running time (ap­

proximately 10 percent difference). Thus the nonconforming elements, both

NC4 and NC8 have been shown to be used satisfactorily for both single and

double curved shells.

5.6 Spherical Shell

One problem which gave a good bit of trouble in the early stages

58

of the development of the finite element procedure was the pressurized shell

of revolution. Therefore a spherical shell with a built in edge and subjected

to a uniform normal load is analyzed. The dimensions of the shell and the ma­

terial properties are given in Fig. 20. In this analysis, nonrectangular ele­

ments are used. Due to the symmetric shape only one quarter of the shell was

actually solved.

This shell has also been solved analytically by Timoshenko [56] and

by finite element method using 24 cubic type isoparametric elements [40J.

Those are a few of the many analyses done for this shell. The latter refer­

ence is of particular interest because it contains the results obtained by

the element with no nonconforming modes, i.e., with the original fully com­

patible element.

The results obtained in this study are shown in Fig. 21. The agree­

ment with the analytical solution and with other finite element solutions are

very good for both meridional bending moment and hoop forces.

Considering the relatively coarse meshes used in this study and the

computational efforts involved in element formulation and solution for twenty

four 60-degree-of-freedom elements used in Ref. 40, the results obtained by

this study are extremely satisfying. The stress resultants obtained by NC4

and NC8 are almost identical except the peak poi,nt where NC4 is slightly un­

favorable.

5.7 Hyperbolic Paraboloid

A hyperbolic paraboloid shell with all edges clamped and subjected

to a uniform normal load is considered next. The geometry and material pro­

perties are shown in Fig. 22. The equation of mid-surface of the shell is

59

z k x y (5.2)

where

k f = a"b

The results obtained by using NC4 and NCB and with two different

corner rises are presented along with the results of other investigators in

Figs. 23 and 24. This shell with the inadvertent variation noted below has

been investigated by Chetty and Tottenham [61J, Pecknold and Schnobrich [34J,

Chu and Schnobrich [lBJ and many others. While the results from Ref. 61 and

Ref. 34 are in good agreement, the results from Ref. lB are quite different

from above two.

Due to a typographical error in the paper of Iyengar and Srinivasin

[63J two different shell geometries have developed for what was thought to

be the same shell. The results presented in Refs. 34 and 61 are for a shell

with corner rise of 1.304 while Ref. 1B used 1.034 inches as listed in Ref. 63.

The results from this study show good agreement to both Refs. 61

and 34 and to Ref. lB, depending on the corner rise used. From this it can

be concluded that all methods would give the same results when applied to the

same problem. Also the results show that while NC4 produces results slightly

higher than the values from the references cited, NCB produced results slightly

lower. The difference is not significant.

5.B Conoidal Shell

The conoidal shells, usually in the truncated form, are most commonly

used for the factory type buildings because of their natural lighting

60

characteristics. Only a limited amount of work has been done on this type

of shell. The more reliable methods of analysis are yet to be developed.

A truncated conoidal shell whose geometry and material properties

are shown in Fig. 25 and subjected to a uniform normal load is investigated.

Two different boundary conditions are considered, i.e., 1) all edges are

clamped and 2) all edges are simply supported. Due to the symmetry, one half

of the shell is actually solved. The equation of the mid-surface of the

shell is

z f = - x L

1 ) (5.3)

This shell was analyzed analytically and experimentally by Hadid

[62J. The results obtained for this study are presented in Figs. 26 thru 29.

5.B. 1 Simply Supported Edges

Figures 26 and 27 show a generally good agreement between the numer-

ical results from Ref. 62 and those of this study. The comparison with the

experimental results reported in Ref. 62 is of particular interest and is very

useful as well because if the analytical result is backed by experimental re-

sults this greatly increases confidence in the numerical solutions.

The difference between results from NC4 (BxB) and NCB (5x4) is

minimal. The mesh sizes are indicated in the parentheses. However, the nor­

mal displacement Wi at X = ~ obtained by NCB is not in such a good agreement

as elsewhere (Fig. 26a). Both results from NC4 and NCB are in closer agreement

with Hadid!s [62].analytical solution than with his experimental solution.

61

5.8.2 Clamped Edges

As with simply supported case, the general agreement between the re-

sults with the nonconforming elements and those from Ref. 62 is reasonably

good. The largest discrepancy among the results is again seen for the normal

I - L ( ) displacement w at x = 2 Fig. 28b. The sensitivity of the structure may be

a major cause for such a discrepancy. For example when the support is clamped

the maximum deflection is reduced by a factor of over five from that value

obtained when the edge was simply supported. This is very unusual behavior

for a shell and is a possible explanation for the large difference between

experimental and numerical results for the simply supported case.

Comparing with the general deflection profile at x = ~. for a simply

supported shell in Fig. 26a, the results from this study seem to be more rea-

sonable than those from Ref. 62 (Fig. 28b). However, the differences shown

in Fig. 28b are not readily explanable in view of the other agreements.

Since relatively coarse grid sizes are used in this study, particular-

ly for NC4, the sharp stress gradients near clamped edge due to the edge dis-

turbance are not adequately represented. In the central region of the shell

where the behavior has settled down to essentially membrane conditions the

agreement is much better.

62

Chapter 6

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDIES

6.1 Conclusions

The development of yet another new finite element was presented in

this study. When the original isoparametric element is used in shell analysis,

either the structural representation is too stiff (lower order element) or too

many unknowns are involved (higher order element) making the use of this type

of element uneconomical. Of the original isoparametric family of shell ele­

ments only the sixty-degree-of-freedom elements, designated as C12 in this

study, can produce reasonable accuracy.

By adding nonconforming displacement modes selectively to the low

order isoparametric elements, highly improved new shell elements have been

established. Addition of various combinations of nonconforming modes to ~rig­

inal elements have been investigated and the elements designated as NC4-4.3R

(linear element) and NC8-4.3R (quadratic element), later abreviated to NC4' and

NC8, respectively, have been found to be most effective in terms of the number

of nonconforming modes used versus the improvement obtained.

For both NC4 and NC8, only two (Nl , N2 in Eq. 3.10 or 3.11) noncon­

forming modes are added to the inplane displacement and four (Nl , N2, N3, N4,

in Eq. 3.10 or 3.11) to out-of-plane displacement. No additional modes are

applied to rotations. These elements, NC4 and NC8 along with C12, have been

tested with deep beam, plate and various shell problems.

The results obtained from these new elements compare with the re­

sults from well established analysis extremely satisfactorily. The improvement

63

incorporated into the four-node element, which has been ignored by virtually

all previous investigators, is most impressive.

The elements NC4 and NCB can be used equally successfully for any

type of shell problems. The element NC4 may be used with more flexibility

for the shells with irregular boundaries and/or irregular shapes. While ele­

ments NCB and C12 can be used advantageously where the sharp stress gradients

exist.

The shells investigated in this study are just a few examples of

what these new elements can solve. The versatility of the elements is evi­

dent. With the unknowns selected for these elements there are no difficulties

involved in adding rib or stiffening elements either concentric or eccentric

to the shell.

6.2 Suggestions for Further Studies

One of the most promising areas which this type of element can be

used effectively is with nonlinear material behavior problems. The extention

to nonlinear material behavior problems is not a difficult task. Unlike other

conventional finite elements, the state of stress or progress of plasticity

can be defined individually at each integrating point in an element. Thus

part of the element may be plastic while the remainder is elastic. Besides,

these types of elements do not have difficulties in obtaining a stiffness

matrix of an element which includes both elastic and plastic regions at the

same time. Therefore, either initial stress approach or updated stiffness

approach can be used in solving the nonlinear problems.

Another extension would be the application to problems of response

64

of shells to dynamic loading. In this case, both material and geometrical

nonlinearities should be considered in order to obtain more accurate solutions.

Very little has been published in this area to date.

Another topic of practical interest is the comparative economies of

reduced integration technique versus nonconforming mode approach. Those two

approaches have some characteristics in common. They take the isoparametric

element as a starting point. They have the same motivations, i.e., to get

rid of the excessive shear energy present in the original low order isopara­

metric element. The comparisons, such as in terms of applicability, gener­

ality and economy, would be of practical importance.

65

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10. Zienkiewicz, O. C. and Cheung, Y. K., "The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs," Proc. Institution of Civil Engineers, August 1964.

11. Gallagher, R. H., "Analysis of Plate and Shell Structures,'1 Proc. Symp. on Application of Finite Element Methods in Civil Engineering, Vanderbilt University, 1969.

12. Bogner, F. K., Fox, R. L. and Schmit, L. A., liThe Generation of Inter­element Compatible Stiffness and Mass Matrices by the Use of Interpola­tion Formulars," Proc. Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pp. 397-444, 1965.

13. Fraeijs de \teubeke, B., "A Conforming Finite Element for Plate Bending," International Journal of Solids and Structures, Vol. 4, pp. 95-108, 1968.

66

14. Bell., Ko .. "A Refined Triangular Plate Bending Finite Element," Inter­national Journal for Numerical Methods in Engineering, Vol. 1, ppm 101-122, 1969.

15. Argyris, J. H., Fried, I. and Scharpf, D., liThe TUBA Family of Plate Element for the Matrix Displacement Method," Aeronautical Journal, Vol. 72, August 1968.

16. Argyri s, J. H. and Scha rpf, D., liThe SHEBA Fami ly of Shell El ements for the Matrix Displacement Method," Aeronautical Journal., October 1968, March, May 1969.

17. Bazeley, G. P. Cheung, Y. K., Irons, B. M. and Zienkiewicz, O. C., IITriangular Elements in Plate Bending - Conforming and Non-Conforming Solutions," Proc. Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, November 1965.

18. Chu, T. C. and Schnobrich, W. C., "Finite Element Analysis of Trans­lational Shells,1J Civil Engineering Studies, Structural Research Series No. 368, University of Illinois, Urbana, Illinois, November 1970.

19. Clough, R. W. and Tocher, J. L., "Finite Element Stiffness Matrices for Analysis of Plate Bending," Proc. Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, ppm 397-444, 1965.

20. Clough, R. W. and Felippa, C. A., "A Refined Quadrilateral Element for Analysis of Plate Bending," Proc. Conference (2nd) on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, Ohio, 1968.

21. Faeijs de Veubeke, B . ., "Displacement and Equilibrium Models in Finite Element Method," Stress Analysis, Edited by Zienkiewicz, O. C. and Hollister, G. S., John Wiley & Sons, 1965.

22. Herrman, L. R., IIA Bending Analysis of Plates," Proc. Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, November 1965.

23. Visser, W., "A Refined Mixed Type Plate 'Bending Element," Technical Note, AIAA Journal, Vol. 7, No.9, ppm 1801-1803, 1969.

24. Herrman, L. R., flFinite Element Bending Analysis for Thin Shells,iI AIAA Journal, Vol. 6, No. 10, pp. 1842-1847, 1968.

25. Prato, C. A., "Shell Finite Element Method via ReissnerDs Principle,1l International Journal of Solids and Structures, Vol. 5, ppm 1119-1133, 1969.

26. Pian, T.H.H .. and Tong, IlRationalizationin Deriving Element Stiffness Ma tri x by Assumed Stress Approach," Proc. Conference (2nd) on r~a tri x Methods in Structural Mechanics, AFFDL-TR-68-l50, Wright-Patterson AFB, Ohio, 1968.

67

27. Connor, J. J. and Will, G. T., IIA t~ixed Finite Element Shallow Shell and Formulation,1I Proc. US-Japan Seminar on Matrix r~ethods of Structural Analysis and Design, 1969.

28. Pian, T.H.H., "Formulation of Finite Element ~-1ethods for Solid Continua," Proc. US~Japan Seminar on Matrix Methods of Structural Analysis and Des i gn , 1969.

29. Tong, P., "New Displacement Hybrid Finite Element Models for Solid Continua,11 International Journal for Numerical Methods in Engineering, Vol. 2, pp. 73-83, 1970.

30. Harvey~ J. W. and Kelsey, S., IITriangular Plate Bending Elements with Enforced Compa t i bi li ty:1 AIAA Journa 1, Vo 1. 9, No.6, pp. 1023-1026, June 1971.

31. Green, B. E., Strome, D. R. and Weikel, R. C., IIApplication of the St i ffness Method to the Ana lys is of Shell Structures:1 ASME, Paper No. 61-Av-58, August 1961.

32. Johnson, C. P. and Clough, R. W., IIA Finite Element Approximation for the Analysis of Thin Shells," International Journal for Solids and Structures, Vol. 4, pp. 43-60, 1968.

33. Connor, J. and Brebbia, C., "Stiffness ~1atrix for Shallow Rectangular Shell Element:1 Proc. ASCE J. of EM Div., No. EM5, October 1967.

34. Pecknold, D. A. and Schnobrich, W. C., "Finite Element Analysis of Skewed Shallow Shells,1I Civil Engineering Studies, Structural Research Series No. 332, University of Illinois, Urbana, 1968.

35. Bogner, F. K.,_ Fox, R. L. and Schmit, L. A., IIA Curved Cylindrical Shell Discrete Element,1! AIAA Journal, Vol. 59 No.4, April 1967.

36. Cantin, G. and Clough, R. W., IIA Curved Cylindrical Shell Finite Element,1! AIAA Journal, Vol. 6, No.6, June 1968.

37. Utku, S., IIStiffness ~~atrices for Thin Triangular Elements of Non-Zero Gaussian Curvature, II AIAA Journal, Vol. 5, No.9, September 1967.

38. Haisler, W. E. and Stricklin, J. A.s IIRigid Body Displacements of Curved Elements in the Analysis of Shells by the Matrix Displacement Method, II Techni cal Note'l AIAA Journal, pp. 1525-1527, Augus t 1967.

39. Ergatoudis, 10:; Irons, B. M. and Zienkiewicz, O. C., "Curved Isopara­metri c, IIQuadri latera 111 El ements for Fi n i te El ement Ana 1 ys is, II I nter­national Journa of Solids and Structures; Vol. 4, pp. 31-42, 1968.

68

40. Ahmad, S., Irons, B. M. and Zienkiewicz, O. C., flAnalysis of Thick and Thin Shell Structures by Curved Finite Elements,fI International Journal for Numerical Methods in Engineering, Vol. 2, pp. 419-451, 1970.

41. Zienkiewicz, O. C., Taylor, R. L. and Too, J. M., "Reduced Integration Technique in General Analysis of Plates and Shells," International Journal for Numerical Methods in Engineering, Vol. 3, ppm 275-290, 1971.

42. Wilson, E. L., Taylor, R. L., Doherty, t~. P. ,and Ghaboussi, J., "Incom­patible Displacement Models," International Symposium on Numerical and Computer t~ethods in Structural Mechanics, University of Illinois, Urbana, September 1971.

43. Gupta, A. K., Mohraz, B. and Schnobrich, W. C., "Elasto-Plastic Analysis of Three Dimensional Structures Using the Isoparametric Element'lfl Civil Engineering Studies, Structural Research Series No. 381, University of Illinois, Urbana, August 1971.

44. Yuzugullu, O. and Schnobrich, W. C., flFinite Element Approach for the Prediction of Inelastic Behavior of Shear Wall-Frame Systems," Civil Engineering Studies, Structural Research Series 386, University of Illinois, Urbana, May 1972.

45. Zienkiewicz, O. C. and Nayak, G. C., "A General Approach to Problems of Plasticity and Large Deformation Using Isoparametric Elements," Proc. Conference (3rd) on Matrix Method in Structural Mechanics, Wright-Patterson AFB, Ohio, October 1971.

46. McNamara, J. F. and ~1arcal, P. V., "Incremental Stiffness Method for Finite Element Analysis of the Nonlinear Dynamic Problems," Proc. Int. Symp. on Numerical and Computer Methods in Structural Mechanics, Uni vers i ty of III i noi s, Urbana, III i noi s, September 1971 .

47. Zienkiewicz, O. C., liThe Finite Element t~ethod in Engineering Science," McGraw-Hill, 1971.

48. Zienkiewicz, O. C., "Isoparametric and Allied Numerically Integrated Elements - A Review," Int. Symp. on Num. and Computer Meth. in Structural Mechanics, University of Illinois, Urbana, September 1971.

49. Doherty, W. P., Wilson, E. L. and Taylor, R. L., "Stress Analysis of Axisymmetric Solids Utilizing Higher_Order Quadril~tefalFinite Elements," SESM Report 69-3, Structural Engineering Laboratory, University of California, Berkeley, California, 1969.

50. Pawsey, S. F., liThe Analysis of Moderately Thick to Thin Shells by the Finite Element Method,1I SESM Report 70-12, Structural Engineering Laboratory, University of California, Berkeley, California, 1970.

51. Popov, E. P., "Introduction to fviechanics of Solids,!' Prentice-Hall, Inc., 1968.

69

52. Bakhrebah, S., IIFinite Element Analysis of Intersecting Cylinders," Ph.D. Thesis, Department of Civil Engineering, University of Illinois, Urbana, 1973.

53. Kaplan, W., IIAdvanced Calculus," Addison-ltJesley.

54. Melosh, R. J. and Banford, R. M., IIEfficient Solution of Load-Deflection Equations,1I Proc. ASCE, Journal of Structural Division, Vol. 95, No. ST4, pp. 661-676, April 1969.

55. Irons, B. r'~., "A Frontal Solution Program for Finite Element Analysis,1I International Journal for Numerical Methods in Engineering, Vol. 2, pp. 5-32, 1970.

56. Timoshenko, S. and L\loinowsky-Krieger, S., IITheory of Plates and Shells:' 2nd Edition, McGraw-Hill, 1959.

57. Scordelis, A. C. and Lo, K. S., IIComputer Analysis of Cylindrical Shells,1I Journal of American Concrete Institute, Vol. 61, No.5, May 1964.

58. Chuang, K. P. and Veletsos, A. S., "A Study of Two Approximate f\1ethods of Analyzing Cylindrical Shell Roofs,1I Civil Engineering Studies, Struc­tural Research Series No. 258, University of Illinois, Urbana, Illinois, October 1962.

59. Abu-Sitta, S. H., "Finite Difference Solutions of the Bending Theory of the Elliptic Paraboloid,1I Bulletin of the International Association for Shell Structures, No. 20, December 1964.

60. Gustafson, W. C. and Schnobrich, W. C., !lMultiple Translatlonal Shells,1I IASS-IMCYC Congreso International Sobre La Applicacion de Estructuras Larninares en Arquitectura, Mexico City, September 1967.

61. Chetty, S.~tK. and Tottenham, H., 'IAn Investigation into the Bending Analysis of Hyperbolic Paraboiloid Shells,iI Indian Concrete Journal, July 1964.

62. Hadid, H. A., nAn Analytical and Experimental Investigation into the Bending Theory of Elastic Conoidal Shell,1I Ph.D. Dissertation University of Southampton, March 1964.

63. Iyengar., K.T.R. and Srinivasin, R. S., IIBending Analysis of Hyperbolic Paraboloid Shell,1I Structural Engineer, Vol. 46, No. 12, pp. 397-401, Decernoer 1968.

64. Tezcan, S. 5., IIComputer Analysis of Plane and Space Structures, II Journal of the Structural Division, Proc. ASCE, Vol. 92, No. ST2, pp. 143-173, April 1966.

Table 1

Element Designations

Nonconforming Modes Applied To DOF per Element Designations -

U,v w a,S Original Condensed

* C4 None None None 20 NC4-2.3 Nl ,N2 Nl ,N2 None 26 20 NC4-2.5 N1,N2 N1,N2 N1 ,N 2 30 20

Linear NC4-4.3 Nl ,N2,N3,N4 Til ,N2,N3,N4 None 32 20 Elements NC4-4.3R** N1,N2 Nl ,N2,N3,N4 None 28 20

NC4-4.5 Nl ,N2,N3,N4 Nl ,N2,N3,N5 Nl ,N2,N3,N4 40 20

-......J NC4-5.5 N1,N2,N3,N4,N5 Nl,N2,N3,N4,N5 N1 ,N2 ,N3 ,N4 ,N5 45 20 a

* C8 None None None 40 NC8-2.5 Nl ,N2 Nl ,N2 N1,N2 50 40

Quadratic NC8-4.3 Nl ,N2,N3,N4 Nl ,N2,N3,N4 None 52 40 Elements NC8-4.3R** Nl ,N2 N1,N2,N3,N4 None 48 40

NC8-5.3 N1,N2,f\f3,N4,N5 Nl,N2,N3,N4,N5 None 55 40 NC8-5.3R** N, ,N2 N1,N2,N3,N4,N5 None 50 40 NC8-4.5. N, ,N2,N3,N4 N, ,N2,N3,N4 N1,N2,N3,N

4 60 40

NC8-5.5 N, ,N2,N3,N4,N5 Nl ' N2 ,N3 ,N4 'I N5 N, ,N 2 ' N 3 ' N 4 ' N 5 65 40 Cubic Element C12* None None None 60 * -'---._---

Equivalent to original isoparametric elements.

**Elements predicted as the most beneficial ones in Section 3.7.

Element & Mesh

C4 (3xl)

NC4-2.3 (3xl)

NC4-4.3 (3x1 )

NC4-4.3R (3x1 ) II (6x2 )

C8 (3xl )

NC8-2.3 (3x1 )

NC8-4.3 (3x1 )

NC8-4.3R (3x1 )

C12 (3xl)

EXACT

---.--

71

Table 2

Comparison of Results of Cantilevered Beam Using Various Elements

(A) Ph ~

tY2 ....-.

Number 6 at A o at B 6 at A Unknowns x 10-5 x 10-1 x 10-5

(B) t P

o at B x 10-3

-----------

12 1 .20 4.00 4.90

12 1 .80 6.00 7.40 --

12 1 .80 6.00 7.80

12 1 .80 6.00 7.40

24 7.58

30 1 .80 6.00 7.59

30 1 .80 6.00 7.61

30 1 .80 6.00 7.61 30 1 .80 6.00 7.61

48 1 .81 6.02 7.68

1 .80 6.00 7.68

----------_ ........

A ~------~----~_r

V:/'

~---~ -IA

~ ____ 3014-----*-

p = 2 1 b

E = 3 x 106 ps i

~ = 0.0

2.02

3.60

3.60

3.60

3.60

3.60

3.60

3.60

3.60

3.65

3.60

72

Table 3

Abscissae and Weight Coefficients of the Gaussian Quadrature Formula

fl PI

f(x) dx = _L llJ~a), -1 J= 1

±a

0-57735 02691 89626

0-77459 66692 41483 0-00000 00000 00000

0-86113 63115 9-l053 0- 33998 10435 S4856

0-90617 98459 38664 0-53846 93101 05683 0-00000 00000 00000

0-93246 95142 03152 0-66120 93864 6()265 0-23861 91860 83197

0-94910 79123 42759 0-74153 11855 9939-l ~40584 51513 77397 0-00000 00000 00000

0-96028 98564 97536 ~79666 64774 13627 0-52553 -24099 16329 0-18343 46424 95650

0-96816 02395 07626 0-83603 11073 2663'6 0-61337 14327 00590 0-32425 34234 O~S09 0-00000 00000 00000

n=2

n=3

n=4

n = 5

n = 6

n = 7

n = 8

n=9

n = 10

H

\-00000 00000 00000

0-55555 55555 55556 0-88888 888~8 81::889

0-34785 48451 37454 0-65214 51548 62546

0-23692 63850 56189 0-47862 1\(1704 993()6 0-56888 88888 88889

0-17132 44923 79170 0- 36076 15730 48139 0-46791 39345 72691

0- J 2948 49661 6S870 0-27970 53914 't,9277 0-381~3 00505 05119 0041795 91836 7346Y-

0-10122 85362 90376 0-22238 1034-t 53374 0-31370 66-f58 77887 0-36268 37833 78362

~OS127 43883 61574 0-18064 8160(J 9-!~57 0-26061 069h4 02935 0- 3123-t 70770 -lO003 0-33023 93550 OI~W

0-97390 65285 17172 0-06667 13.+43 086SS 0-86506 336(16 SSl)S5 O-I-l9.t5 134\) I 5!l5S 1 0-67940 956S2 9t)()~-1- 0-21lJOS 63«25 15t)S2 0-43339 53\)41 29247 Q-2c1'J2() 671l)3 Ol)996 0-14SS7 43JS9 S 16~ I Q-2lJ552 42247 14753

73

(a) Linear Element

(b) Quadratic Element

(d) Quadratic Element--Nodes Across Thickness Eliminated

(c) Cubic Element

(e) Cubic Element--Nodes Across Thickness Eliminated

FIG. 1 3-DIMENSIONAL ISOPARAMETRIC ELEMENTS

74

n

M M

(a) Linear Element Under Pure Bending

or or

(b) Constrained Deformation

I L _________ .....J

(c) Actual Deformation

FIG. 2 LINEAR ISOPARAMETRIC ELEMENT UNDER PURE BENDING

75

Conforming Mode

Nonconforming Mode

II

Combined Mode ,. I

/ I I L.... ___________ J

FIG. 3 RESTORING THE ACTUAL BEHAVIOR OF THE ELEMENT BY ADDING NDNCONFORr~ING MODE

76

(a) Displacement Components at the Top and Bottom Nodes

z,w

Y,V

Node i

(b) Global Coordinates, Local Coordinates and Corresponding Displacement Components

FIG. 4 DEFINITIONS OF GLOBAL AND LOCAL COORDINATES

77

(a) N]

(b) ~

FIG. 5 NONCONFORMING DISPLACEMENT (w) MODES FOR LINEAR ELEMENT

78

(b) R2

79

FIG. 7 INTEGRATION POINTS FOR PLATE (n = 3)

.33998

0.86113

FIG. 8 GAUSS INTEGRATIONS

80

a/2 = t = E =

a/2 ]J =

x

FIG. 9 TYPICAL UNIFORM MESH (2x2) OF CLAMPED SQUARE PLATE UNDER CONCENTRATED LOAD AT THE CENTER

40"

1 "

3 x 0.3

~ 100 ...

c o .,.... +-> ~

..--o

C/)

..--ro ClJ

c.c::

4-o OJ en ro

+-> c OJ u s­OJ

0..

80

60

40

20

a 1 xl

8-Node Element (C8)

". 4-Node El ement (C4).......""

?v? C-AC- 3x3

Mesh Size in 1/4 of plate

~

4x4 5x5

FIG. 10 CENTRAL DEFLECTION OF CLAMPED SQUARE PLATE UNDER CONCENTRATED LOAD AT THE CENTER

106 psi

81 260

NC4-4.5

~

In

120 c: 0

or-..j...J

:::s 100 0

(,/')

ro 80 OJ 0:::

4-0 NC4-4.3R OJ 60 O"l ro

+-l c: OJ u 40 $.... OJ

0...

20

0 2x2 4x4 6x6 8x8

(a) Linear Element

120 NC8-5.5

~ Y .. ... 100 c: ~~ NC8-4.3R 0

NC8-5.3R or-+-l :::s

r-- 80 0

(,/')

ro 60 OJ

0:::

4-0

OJ 40 O"l ro

+-l c: OJ u 20 $.... OJ

0...

0 1 xl 2x2 3x3 4x4 5x5

Mesh Size in 1/4 of Plate (b) Quadratic and Cubic Elements

FIG. 11 CENTRAL DEFLECTION OF CLAMPED SQUARE PLP,TE

r-ct5 OJ

cr:: 4-o OJ en ct5 +' s:: OJ u S­OJ

0..

100

80

I I

I I

I I I

82

ACM

" " f1 ,

......... -- ........ '"",,-~ ........ '" / ...-..., ...............

/C12 I

I

NC4

2 4 8

Mesh Size in 1/4 of Plate 10

Meaning of Symbols

M, AC~l

LCCT-9 DVS Q-19

Incompatible but complete triangular elements; 12 external OOF (Ref. 9)

Two LCCT-9 triangles; 12 external DOF (Ref. 19) Compatible quadrilateral; 16 external DOF (Ref. 13) Four LCCT-11 triangles; 12 external and 7 internal DOF (Ref. 20)

FIG. 12 COMPARISON OF CENTRAL DEFLECTION FROM VARIOUS PLATE BENDING ELEMENTS

83

80 ~

... c 0 y +-' t :::::i .--0

(/) 60 r 10 il

.--rd OJ t = 111

0::::

4- E = 3 x 106 psi 0

OJ 11 0.0 0') rd 40 ~x +-' C

I 4 ~ I OJ u S- r OJ

0..

4 8 12 16

Number of Elements

FIG. 13 CLAMPED EDGE CIRCULAR PLATE UNDER UNIFORM LOAD

z

(a)

84

L = 372"

R = 372"

t = 3.75 11

a = 40° E = 3 x .06 psi

II == 0.0

\ Case I ,/

\ / \ Case II ,/

\ /

FIG. 14 GEOMETRY AND MATERIAL PROPERTIES OF CIRCULAR CYLINDRICAL SHELL

-10

10

VI 20 OJ

..c:: u c

3: 30

40

20

~ Scordelis & Lo (Ref. 57)

" NC4} Loadi ng Case I • NCB

t::. NC4} Loading Case II o NCB

o C12 II CB }

Ahmad (Ref. 40)

(a) Wi at Midspan

cp, degrees

30

FIG. 15- DISPLACEMENTS OF CIRCULAR CYLINDRICAL SHELL UNDER UNIFORM LOAD

40

Vl QJ

..c u I::

N i o

.. >

Vl QJ

..c u

12

2

1::4 ...-

I o

8

o

85

------ Chuang (Ref. 58)

6 NC4} loading Case I • NC8

6 NC4} loading Case II o NC8

~ ~~ 2 . }. 'Ahmad (Ref. 40) .

---- Exact

(b) v at End

10 20

<p, degrees

(e) u' at Midspan

FIG. 15 (CDr.JTH1UED)

30 40

20

-+-l '+-....... (/)

a. ...... ..;,t.

">, 40 :z

60

-4

" -+-l 4-.......

(/)

a. -2 .,... ..;,t.

.. -x z

0

" ~

86

-- Scordelis & Lo (Ref. 57)

A NC4 } Loading Case I /I NC8 6- NC4 } Loadi ng Case II 0 NC8 0 C12 } Ahmad (Ref. 40) II C8

(a) N at Midspan y

" ~ 0

(b) N I at Midspan x

<1>. degrees

FIG. 16 STRESS RESULTANTS OF CIRCULAR CYLINDRICAL SHELL UNDER UNIFORM LOAD

87

3.0 Scorde1is & Lo (Ref. 5?)

" NC4 Loading Case • " • NC8 •

" {j, NC4 Loading Case II 0 NC8

2. 0 C12

fIIB C8 } Ref. 40 +l 4- Exact .......... +-l 4-

I VI fIIB 0-

'r-.:>L.

X 1.0 ::E:

(c) M I x at Midspan

LO

+-l 4-.......... +-l 0.5 4-

I VI 0-

..:;;:

>, 0 ::E

-0.5 0 10 20 30 40

¢, degrees (d) r~ at r,l ids pan

y

FIG. 16 (CONTINUED)

10

8

;t: 6 ""­til Cl..

>,

x z 4

2

o

+-> ~ ......... +-> ~

I til Cl..

'r-.::.t.

>,

X ~

88

Scorde1is & Lo (Ref. 57)

It. NC4 } Loading Case I

• NC8

tJ. NC4 } Loadi ng Case II 0 NC8

(e) N I I at End x y

2r------------r------------r-----------~-------------o C12 I C8 ;} Ref. 40

I

10 20 30 40

cp, degrees

(f). Mx'y' at End

FIG. 16 (CONTINUED)

(/)

QJ ...c u s::

N I o

89

6 E = 3 x 10 psi ]J = 0.0 t = 0.429 11

q = 104 lb/ft2

a = b = 48 11

FIG. 17 GEOMETRY AND MATERIAL PROPERTIES OF A SQUARE ELLIPTIC PARABOLOID

x a

--- Chu & Schnobrich (Ref. 18) Gustafson & Schnobrick (Ref. 60)

6 NC4} This study o NC8

o

w Along Center Line

FIG. 18 VERTICAL DISPLACEMENT OF SIMPLY SUPPORTED SQUARE ELLIPTIC PARABOLOID UNDER UNIFORM LOAD

1 .0

100

75 t::

'r-

'-..0 ,.......

>, 50 z;, x z

-150

-100 t:: 'r-

'-..0

>,

x -50 z

3/4 'L b

90

1/2

(a) Nx" Ny' Along the Center Line

-~-Chu & Schnobrich

__ Gustafson & Schnobrich

o N I

II N x I } NCB y } This study

tJ. NX' NC4 ,. N I }

Y

1/4 a

O~~ ____ , ____ L-__________ L-__________ ~ ________ ~

3/4 1.0 a 1/4 1/2

(b) N I I Along the Edge x y

x a

FIG. 19 STRESS RESULTANTS OF SIMPLY SUPPORTED SQUARE ELLIPTIC- PARABOLOID UNDER UNIFORM LOAD

~ I+--~ l+-

I VI 0.. ..... ~ .. -

>. :::E

0

2

4

1.0

91

3/4

(c) M I Along Center Line y

1/2

FIG. 19 (CONTHJIJED)

q

1/4

R = 90"

t = 3"

0

11 = 0.1333

E = 3 x 106 psi q = 1 psi

FIG. 20 GEm1ETRY MJO r~ATERIi\L PROPERTIES DF SPHERIC/\L SHELL

o

s::: .r-......... ..Cl ..-

I

s:::

-& :::E:

C 'r-......... ..Cl ..-

'" <D z:

92

40

Analytical (Ref. 56) 30 0 C12 (Ref. 47)

t:. NC4 } This study 0 NCB

20

10

0

(a) Meridional Bending Moment

O~----~------~------r-----~------r-------r-----i

10

20

30

40

o o

50 ~----~------~----~------~------~----~-------35 30 25 20 15 10

cp, degrees

(b) Hoop Force

FIG. 21 STRESS RESULTANTS OF SPHERICAL SHELL UNDER UNIFORM LOAD

5 o

0

-4.0

U1 OJ

.J:: u

-8.0 C or-

M I 0 ..-

~

-12.0

-16.0 o

93

z

a = b = 6.46"

t = 0.25 11

f = 1.304 11

or 1 .034 11

Jl = 0.39 E = 5 x 105 psi

q = 1.0 psi

FIG. 22 GEOMETRY AND MATERIAL PROPERTIES OF HYPERBOLIC PARABOLOID

Chetty & Tottenham -,,- Pecknold & Schnobrich

--- Chu & Schnobrich

6. NC4} w/f = 1.304 0 NC8

-::?" .l NC4} w/J = ~ ... II NC8 1 3 .. ~d' .. :;;--

e......-t:r'''--

/' /'

/'

1/4

~/

x a

1/2 3/4 1.0

FIG. 23 . VERTICAL DEFLECTION ACROSS MIDSPAN OF tLAMPED HYPERBOLIC PARABOLOID UNDER UNIFORM NORMAL LOAD

>,

x z

c:

'-. .0

..-: c:

'::£.>,

94

-2.0------------~----------~------------~----------~

-1. 0

0

1.0

2.0

3.0 o

(a) N I I Across Midspan x y

Chetty & Tottenham _ .. - Pecknold & Schnobrich

---t::. NC4

0 NCB

At. NC4

• NCB

Chu & Schnobrich

w/ f = 1.304

w/ f = 1.034

1/4 x a

(b) M I Across Midsection y

}

1/2

This Study

3/4 1.0

FIG. 24 STRESS RESULTANTS OF CLAMPED HYPERBOLIC PARABOLOID UNDER UNIFORM NORMAL LOAD

95

x

b = 49.5"

t = 0.5"

L = 72" h = 18" 1 h = 9"

2 -6 E = 5.62 x 10 psi

~ = 0.15 q = 60 pst

(pressure)

FIG. 25 GEOMETRY AND MATERIAL PROPERTIES OF CONOIDAL SHELL

-5

FIG. 26 DISPLACEMENTS OF SIMPLY SUPPORTED CONOIDAL SHELL UNDER UNIFORM LOAD

U'l QJ .c u s::

('Y) I 0

3:

U'l QJ .c u s::

('Y) I o

-10

-5

o

\

96

--- H-~leth } Ref. 62

- -0-- Exp

6 NC4 } This Study o NC8

I

(b) w at y = 0

1/4 1/2 3/4

\ / \ /

5 \ / \ ~

\ 6 I \ I \ I ~ I \ I \ ;1 \ I

\ / \ / b /

, n

10

15

" //"" " ' 0--""/ '0 __ -,

20 0 1/4 -

x 1/2 3/4 [

(c) I b

w at y = z

FIG. 26 (CONTINUED)

1.0

1.0

+-l 4--V')

0.. 'r-..::..:. .. -x z

+-l 4--V')

0.. or-~

'" -x z

-0.6

-0.4

-0.2

;..;0.4

-0.2

____ H-Meth

--0-- Exp

97

} Ref. 62

A NC4

o NCB } This Study

(a) N I at y = ~ x z

- l (b) Nx' at x = Z

Y.. b

FIG. 27. STRESS RESULTANTS OF SIMPLY SUPPORTED CONOIDAL SHELL UNDER UNIFORM LOAD

0.6

0.4

+-> 4--.. (/)

0.. .~

..:.t:.

>, 0.2

z

98

~o

-, '0""

....... -.0....... 0 -------0.... IJ ,

H-Meth } Ref. 62 - -cr- -- Exp

t:. NC4

o NC8 This Study

\ \ \ ~ \ \ \ \ \

-0.2~ __________ ~ __________ ~ __________ ~ __________ ~

o 1/4 -x 1/2 3/4 1.0 [

(c) Ny' at y = 0

0.4

.~

..:.t:.

" 0.2

0 0 1/4

Y 1/2 3/4

b

(d) N • - L at x = -y z

FIG. 27 (CONTINUED)

.,...

........

..0 ..-c: .....

c: .,... ........ ..0 ..-

0

c ..... .. -

:£:."»>

1/4

a

4.0

8.0

99

........ , " ,

"

H-Meth

" ~ \

6 \ \ \ \

\ } Ref. 62 --0-- Exp

6 NC4 } This Study o NCB

- 1/2 x l

3/4

12.0 L-.. ____ --L. _____ ...L-.. ____ .........;;:m:;;;...... ___ ----'!

o 1/4 1.. 1/2 3/4 1.0 b

(f) My' at x = ~

FIG: 27 (COnTINUED)

til Q)

..c u 1.0 s::

(V) I 0

~

2.0

Vl

~ 1.0 u s:: .....

(V) I o

.. ') n L.V

o

100

-x 1/4

[ 1/2 3/4

-- Ref. 62

t:. NC4 }This study 0 NC8 0 0

t:.

I

(a) w at y = 0

I - L (b) w at x = "2

FIG. 28 DEFLECTIONS OF CLAMPED CONOIDAL SHELL UNDER UNIFORM LOAD

1.0

1.0

t:. 0 o

-10.0

. -20.0 c:: '''' ........ ..a ,....

.. 40.0

101

KKTG } Ref. 62 --0--- Exp

t:. NC4} Thi s Study o NCB

o 0 / --a--:;!{1

I /0

-50.0 L ____ ~~::::......---~----__:~----~ o - 1/2 3/4 1.0 x

[

(b) N I at y = 0 y .

FIG. 29 STRESS RESULTA~~TS OF CLA~'1PED COrJOIDAL SHELL UnDER UN I FORr'1 PRESSURE

s:: or-......... ..0

-6.0

102

Ref. 62

A NC4 } This Study o NCB

1/4

(c) M I at y = 0 x

_ 1/2 x [

FIG. 29 (CONTINUED)

3/4 1.0

103

APPENDIX A

SHAPE FUNCTIONS

If a shell is defined by the mid-surface coordinates only, the

shape functions for the shell may also be represented, as two dimensional

elements, using ~ and n only. Here ~ and n are two curvilinear coordinates

in the mid-surface of shell which vary between -1 and 1 (Fig. 1).

A.l Linear Element

N. 1

1 = - (1

4 + ~) (1 o

+ n) o

with ~ = ~ E,. and n n n· o 1 0 1

A.2 Quadratic Element

Corner Nodes

Mid-side nodes (~. = ± 1, n. = 0) 1 1

AD3 Cubic Element

Corner nodes

+ ~) (1 o 2

- n)

N .. = -3'2 (' + ~ ) (1 + n ){-10 + 9(~2 + n2)} 1 0 0

(A.l )

(A.2)

(A.3)

(A.4)

104

Mid-side nodes (~ =

(A.5)

l05

APPENDIX B

LOCAL COORDINATE SYSTEM

Besides the basic two coordinate systems, the Cartesian and curvi-I I

linear coordinates systems, another orthogonal coordinate system, x , y , and I

Z , is defined locally (Fig. 4). The corresponding unit vectors, vl' v2' and

v3'" in the new directions of new coordinate systems are also shown in Fig. 4.

The unit vector normal to the shell surface can be established from

the cross product of two vectors tangential to the reference surface. This

has been done very efficiently by taking the first tltJO columns of the transpose

of the Jacobian which is obtained in the process of forming the stiffness

matrix.

The transpose of the J~cobian is

X'l; x, n X, I;:

[J] T = Y'l; y'n y., I;: (B. 1 )

z'l; Z, n z'l;:

Taking the first two columns

x'l; x, n

V3 = Y'l; x Y'n (B.2)

z'l; z, n

Then normalizing V':j ') one can get unit vector v') . ..J -.J

Once the vector v3 is defined it is possible to erect a infinite

number of mutua lly perpendicular vectors orthogonal to it. It is convenient

106

to choose axes so as to relate to the global axes, x or y, wherever possible.

First it is checked if the first or second term in vector v3 is zero. If tHe

second term is zero, a unit vector along the y-axis, denoted as (j), is es-

tablished. If the second term is non-zero, a unit vector along the x-axis,

(i), is established.

For the former case

j x (B.3)

Now, vl is a unit vector perpendicular to a plane defined by the y-axis and

v3° v2 is erected silTljJly by

x (B.4)

For the latter case the calcul~tion_begins with v2

instead of vl

in Eq. B.3 and follows the same procedures as the former case.

= x i (8.5)

The direction cosine matrix can be expressed by these three unit vectors.

[e] (B.6)

Thus the local I I

coordinates x , y , z and correcponding direction cosines

vl ' v2' v3 are obtained. It is noted here that even though new coordinates I

X ,y lie in the surface defined by ~, n they do not coincide with ~, n gen-I I

erally. For the plate, the directions of x ,y z are"coincide with the di-

rections of global axes, x, y, z.

107

APPENDIX C

LOCAL STRAINS AND TRANSFORMATIONS

At first, the global derivative matrix is transformed into the local

derivative matrix by the standard axis rotations.

I J I

U, I x v, I

x W, I

x U, x V, x w, X

I I

[eJT [OJ (C.l ) u, I v, I W, I = U 'y V'y W'y Y Y Y I I I U, I z V, I z W, J z U, z v, z W, z

3u I dU where I

etc. u, I, u, etc. --, , dX x x 3x

[eJ = direction cosine matrix [vl ' v2' v3 J

Since the shape functions, N. and N. in Eq. 3.8 and Eq. 3.12, are 1 J

defined with respect to the curvilinear coordinates ~, nand r;, (Appendix A)

another transformation from the curvilinear to global coordinates is needed

in order to evaluate the global derivatives of displacement with respect to

global coordinates. By applying the chain rule [53J and introducing the in-

verse of the Jacobian, the global derivative matrix is expressed by

u, x v, x w, x u,~ v'E, w,~

U'y v, w, = [J] -1 u, v, w, y Y n n n (C.2)

u, z v, z

w, z u'r;, v'(, w'r;,

where [JJ -1 is the inverse of Jacobian

108

x,(; y, E~ z'(;

[JJ = x, Y'n z, . (C.2.1 ) n n

x'L: Y , L: z'L:

Substituting Eq. C.2 into Eq. C. 1

I I , U, I x V, I x

W, I x u'l: v'l: w'l: I I I [e]T[J]-l [eJ u, I V, I W,I = U, v, W, Y Y Y n n n

I I I

U, I z V, : z

W, I z u'L: v'L: It! , L:

(C.3)

Now, it is possible to evaluate the local displacement derivatives numerically

for any given set of curvilinear coordinates (l:,n,L:).

Taking the'advantage of the normality condition of vectors in cal-

[ T -1 culating 8J [J] ,one may get a matrix [A] in a special form

o

[A] [e]T[J]-l o (C.4)

Introducing new expressions

B1 . = AllNi,r, + A12 N. 1

,., n

B2. = A21 N;,l: + A22 N. 1

1 ,n (~.5)

B3. = A33 Ni,l: 1

aN. where N. l:' etc.

_ 1 etc.

1 , -~ ,

109

Picking up proper terms from Eq. C.3 and utilizing Eq. 3.12, the strain vector

can be obtained

I T [: C. oT [:1 e T [~ + C. 8

T Gt {c } B. 8 + + B. 1 1 1 J

w . J

(C.6)

where

B. Bl 0 0 1

0 Bl 0

B2 81 0

0 0 Bl

0 0 . B2

C. = c;B l ;0 0

1

0 c;B2 0

c;B 2 cB :> 1 0

B3 0 c;B l

0 B3 c;B2

and B. and C. are defined in a similar way. J J

The approaches in this appendix are basically the same as given in

Ref. 40 except minor changes.

110

APPENDIX 0

NUMERICAL INTEGRATIONS

With isoparametric element analysis, complex calculations are in-

volved in determining both the element stiffness and the load vector. Numer-

ical integration is the only practical way to perform the calculations.

Gaussian integration is most suitably used for these computations.

For simplicity, one dimensional function f(~) is considered. First,

sampling points, so located to achieve best accuracy, should be determined

(Fig. 8). Then, with corresponding weighting coefficients H., the integrals 1

can be written for n sampling points as

I

1

J -1

f(C:) de: = n I H. f(e:.) 1 1 1

(D. 1 )

The locations of sampling points and the weighting coefficients are

given in Table 3 which was taken from Ref. 47.

For two and three dimensional functions, the integrals can be

written respectively, as

1 1 n m

J (

I J

F(e:,n) de: dn = L .L H.H.f(e:., n·) (0.2) i = 1 j=l 1 J 1 J

-1 -1

and 1 1 1 1 ( ( ( n m

I j j ) f(e:,n,~) d~ dn d~ = ·I L L H.H.H k (~i ,nj ,sk) i=l j=l . k=l • 1 J

-1 -1 -1 (0.3)

Frequently n, m arid 1 are taken as the same number for convenience.

The minimum size of integration mesh should be so determined that

111

the volume of the element is integrated adequately. The integration meshes

of 2x2, 3x3 and 4x4 are sufficient for linear, quadratic and cubic elements

respectively.

112

APPENDIX E

TYPICAL COMPONENTS OF ELEMENT STIFFNESS MATRIX

The element stiffness matrix Ke in Eq. 3.20 can be partitioned as

( E . 1 )

where Kll , K12 and K22 are defined in Eqs. 3.20.1 to 3.

The matrix Kll can also be subdivided into n x n submatrices for

an element with n nodes.

Kll r K K K .. K. 1 ----11 --12 --1 J -1 n

K ---21

j (E.2)

K' l K .. K. -1 -lJ -In

K K . K --n 1 nJ nn

whose typical submatrix, K .. , is of the order of 5 x 5 reflecting the 5 degrees -lJ

of freedom at each node.

where

The matrix K .. is further subdivided as -1 J

K .. K K -lJ -pp ---pr

(3x3) (3x2)

K K ---rp -rr (2x3) (2x2)

1 1 1

J r r T K J J

8 [B . DB . ] 8 I J I dE: dn dZ;; --pp 1 J -1 -1 -1

etc.

(E. 3)

113

If the thin shell assumptions are made, where the variation of [8J

with respect to ~ is negligible, an explicit integration with respect to C;, (1

i.e., in the form of J f(C;) dr::;, can be performed easily_ By doing -1

so, the

numerical integration is needed in only two directions (~,n) and the computa-

tional efforts for element stiffness matrix can be reduced substantially.

Thus,

where

K -pp

K -rp

K "~pr

1 (

= 2 J -1

ti {

-1 (1

tj J -1

t.t. --' ~

6

1

J -1

{ -1 (1

J -1

(1

J -1

t.t. J ' J + -2--

1

-1

T 8[B.08.J e IJI d~ dn 1 J

¢~ 8[C.OB.J 8T IJI d~ dn 1 1 J

e[B.DC.J 8T ¢. IJI d~ dn

1 J J

(1 T 'T j ¢. 8[B.OB.J 0 ¢. IJI d~ dn

'1 1 J J -1 1

J( ¢ ~ e [C. DC . ] 8 T ¢. I J I d~ dn

1 1 J J -1

[B.DB.J , J B1i011Blj + B2i023B2j Bli012~2j + B2i033Blj

(E.3.1 )

o

B2i012Blj + Bli033B2j B2i022B2j + Bli033Blj' 0

o

o

o

o Bli044Blj

+ B2iOSSB2j

(E.3.2)

114

[C.DB.] :::; 0 0 B3i D44Blj 1 J

0 0 B3i D55B2j

0 0 0

[C.DC.] r

B3iD44B3j 0 0 1 J

0 B3i D55B3j 0

0 0 0

The submatrices K12 and K22 in Eq. E.l are, respectively, in the forms of

K .. r " "'1 --1 J

l:PP j --rp

(E.4)

and

K .. [~p] --1 J (E. 5)

The terms in these matrices can be obtained in the similar way as Eq. E.3.1

and Eq. E.3.2 by simply changing B., etc. to ~. etc. 1 1

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7

Spell rit \" Cl a s sifica tion

DOCUMENT CONTROL DATA - R&D I 5"clJrity cla!>sificatiotl of title, body of abstract ilnd indexing annotation must be erl'tered when the overall report is classified) t---~---....:..-......~-----.:.:....---""""""""T""----"':'-----"-;"-"""""""

C" ·.~IN"'TING ACTIVITY (Corporate allthor) 2<1. REPORT SECURITY CLASSIFICATION

Department of Civil Engineering Univers i ty of III inois Urbana, Illinois 61801

2h. GROUP

\ f-. L P 0 R TTl T L E

Use of Nonconforming Modes in Finite Element Analysis of Plates and Shells

·1 (("iCRIPTIVE NOTES (Type of report and. inclusive Jates)

:, AU THOR(S) (First name, middle initial, last name)

Chang-Koon Choi and W. C. Schnobrich

6 REPORT DATE 7B. TOTAL NO. OF PAGES 17b. NO. OF REFS

September 1973 120 Ila. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S)

N00014-67-A-0305-00l0 SRS 401 b. PROJECT NO.

c.

d.

9b. OTHER REPORT NO(S) (Any other numbers that may be Bssigned this report)

U,I LU-ENG-73-20l9 10 DISTRIBUTION STATEMENT

Release Unlimited

11. SUPPLEMENTARY NOTES

13. ABSTRACT

1~. SPONSORING MILITARY ACTIVITY

Office of Naval Research Structural Mechanics Branch Arlington, Va. 22217

The development of yet another finite element is presented in this study. By adding nonconforming displacement modes selectively to the low order isoparametri elements, highly improved new shell elements are established. Addition of various combinations of nonconforming modes to the original elements has been investigated and the elements, both linear and quadratic, which have two noncon­forming modes added to the inplane displacements, four to the out-of-p1ane dis­placement and no additional modes to rotations have been found to be most effective in terms of the number of nonconforming modes used versus the improve­ment obtained. These elements have been tested with deep beam, plate, and various shell problems. Cylindrical, elliptic paraboloid, spherical and conoidal shells were investi~ated in order to demonstrate the versatil ity of newly de­veloped nonconforming elements. The results obtained from these investigations compare favorably with the results from well-establ ished analyses. The improve­ment incorporated into the four-node element is most impressive.

D FORM -1 NOV651

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73 SIN 0101-807-6811 Security Classification

A-3140B

j" LIN K A LIN K B LIN K C

KEY WORDS

ROLE WT ROLE WT ROLE WT

~ Structural She 11 s Numerical Analysis Finite Element

I

I

FORM 1 1 NOV 65

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Security Classification ·'\-31409