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 Order Number 8916526

A nonlinear finite element for curved beams

Suhendro, Bambang, Ph.D.

Michigan State University, 1989

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UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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 A NONLINEAR FINITE ELEMENT FOR CURVED BEAMS

By

Bambang Suhendro

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A DISSERTATION
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Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1989
 ABSTRACT

A NONLINEAR FINITE ELEMENT FOR CURVED BEAMS

By

Bambang Suhendro

A procedure for the computation of nonlinear elastic response of

curved beam structures is presented. The structure is represented by

beam finite elements curved in one plane but deformable in three

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dimensional space. The curved axis of the element is represented by a

second order polynomial in the curvilinear coordinates. Geometric

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nonlinerities are considered by including the effect of rotations on

the longitudinal strains. In deriving the linear stiffness matrix, the

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displacement functions are approximated by cubic polynomials. However,

the incremental (or nonlinear) stiffness matrices are derived by

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assuming that the longitudinal displacements are interpolated by linear

polynomials while the interpolations for the other displacements remain

unchanged. The nonlinear terms in the strain expression are averaged

over the element length. Differentiation twice of the strain energy

expression yields the linear stiffness matrix ,[k] , and the first and

second order incremental stiffness matrices ,[nl] and [n2], of the

element.

Assuming that the system is elastic and conservative, the

equilibrium equation is obtained from the first variation of the

potential energy. The problem is solved by the Newton-Raphson method

using load increments.

 Bambang Suhendro

A computer program was prepared for the implementation of the

nonlinear equilibrium solution. Numerical results were obtained

involving arches with in-plane and out-of-plane behavior. Various types

of geometry, loading, and support condition were considered.

Numerical results indicated that the proposed method, which is

based on a fixed Lagrangian coordinate system, works very well for

"small displacement problems" ( 2% or less of the arch span) as well

as for 11 intermediate displacement problems" (2-25% of the arch span).

For all of the numerical problems considered, accurate load-

deflection curve may be obtained by using at most eight elements to

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represent the entire arch. For symmetrical problems, only one half of

the arch (four elements) need be considered. Many cases required only
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two elements.

Comparisons of numerical results with those of other methods

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indicate that the method presented is more accurate and effective than

the others.
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The solution procedure based on an updated Lagrangian coordinate

system is also presented. The procedure is necessary if large

displacements (say 25% or more of the arch span) are involved.

In addition to the displacement response, the response of stresses

was also investigated. Furthermore, amplification factors for

displacement and stresses were studied. The result indicated that the

displacement amplification factor was always larger than the stress

amplification factor.
 ACKNOWLEDGMENTS

The writer wishes to express his appreciation to his major

professor, Dr. Robert K. Wen, professor of Civil Engineering, for his

guidance, encouragement, and numerous helpfull suggestions during the

conducting of the research and preparation of this dissertation. Thanks

also to members of the writer's doctoral committee : Dr. P. Soroushian,

professor of Civil Engineering, Dr. Nicholas J. Altiero, professor of

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Mechanics and Materials Science, and Dr. Chi Y. Lo, professor of
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Mathematics, for their valuable suggestions.

Appreciation is also due to the Department of Civil Engineering,

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Faculty of Engineering, Gadjah Mada University, and the government of

Indonesia for providing me with a scholarship through MUCIA-World Bank

IX Education Project to support my graduate studies at Michigan State

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University.

Special appreciation is due to his wife, Magda Bhinnety, son,

Indranu Sulistyo Atmoko, and daughter, Indrati Sudewi, for their

understanding, patience, and cooperation. Appreciation is also due to

his parents, Soedjoed Siswosoedarmo and Suyati, for their prayer and

encouragement.

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 TABLE OF CONTENTS

Page

ACKNOWLEDGMENT iii

LIST OF TABLES vii

LIST OF FIGURES viii

CHAPTER

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I. INTRODUCTION 1

1.1 GENERAL 1
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1.2 OBJECTIVE AND SCOPE 2
1.3 LITERATURE REVIEW 7
1.3.1 LINEAR EQUILIBRIUM 7
1.3.2 BUCKLING ANALYSIS 7
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1.3.3 NONLINEAR EQUILIBRIUM ANALYSIS 8

1.4 NOTATION 11

II. FINITE ELEMENT MODEL FOR A CURVED BEAM 15

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2.1 GENERAL 15
2.2 STRAIN-DISPLACEMENT RELATION 15
2.3 STRAIN ENERGY EXPRESSION 17
2.4 FINITE ELEMENT FORMULATION 18
2.4.1 DEFINITION OF COORDINATE SYSTEMS 19
2.4.2 ELEMENT GEOMETRY 19
2.4.3 DISPLACEMENT FUNCTIONS 22
2.4.4 ELEMENT STRAIN ENERGY AND STIFFNESS MATRICES 24
2.4.4.1 QUARTIC AXIAL STRAIN MODEL 25
2.4.4.1.1 LINEAR STIFFNESS MATRIX 27
2.4.4.1.2 INCREMENTAL STIFFNESS
MATRICES 28
2.4.4.2 AVERAGE AXIAL STRAIN MODEL 30
2.4.4.2.1 LINEAR STIFFNESS MATRIX 31
2.4.4.2.2 INCREMENTAL STIFFNESS
. MATRICES 31
2.4.5 EQUILIBRIUM EQUATIONS 32

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CHAPTER Page

III. NONLINEAR EQUILIBRIUM ANALYSIS OF CURVED BEAM STRUCTURES 34

3.1 GENERAL 34
3.2 NEWTON-RAPHSON METHOD 34
3.2.1 CONCEPT 34
3.2.2 NEWTON-RAPHSON METHOD FOR FIXED COORDINATES 36
3.2.3 CONVERGENCE CRITERION 38
3.2.4 STRESS COMPUTATION 38
3.3 COMPUTER PROGRAM 39

IV. NUMERICAL RESULTS 42

4.1 GENERAL 42
4.2 LINEAR EQUILIBRIUM PROBLEMS 43
4.2.1 CONCENTRATED INPLANE LOAD AT CROWN 43
4.2.2 CONCENTRATED OUT-OF-PLANE (TRANSVERSE) LOAD
AT CROWN 44

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4.3 NONLINEAR LOAD-DISPLACEMENT BEHAVIOR FOR
SMALL DISPLACEMENT PROBLEMS 44
4.3.1 INPLANE PROBLEMS 45
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4.3.1.1 A 90°-HINGED CIRCULAR ARCH SUBJECTED
TO UNIFORM RADIAL LOAD 45
4.3.1.2 A HINGED PARABOLIC ARCH SUBJECTED TO
UNIFORM LOAD ON HORIZONTAL PROJECTION 46
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4.3.2 OUT-OF-PLANE PROBLEMS 46

4.3.2.1 A 90°-HINGED CIRCULAR ARCH SUBJECTED
TO UNIFORM RADIAL LOAD 46
4.3.2.2 A HINGED PARABOLIC ARCH SUBJECTED TO
UNIFORM LOAD ON HORIZONTAL PROJECTION 47
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4.4 NONLINEAR LOAD-DISPLACEMENT BEHAVIOR FOR

INTERMEDIATE DISPLACEMENT PROBLEMS 48
4.4.1 A 28°-CLAMPED CIRCULAR ARCH SUBJECTED TO
A VERTICAL CONCENTRATED LOAD AT CROWN 48
4.4.2 A 60°-CLAMPED CIRCULAR ARCH SUBJECTED TO
A VERTICAL CONCENTRATED LOAD AT CROWN 49
4.4.3 A 60°-CLAMPED CIRCULAR ARCH SUBJECTED TO
A SKEW CONCENTRATED LOAD AT CROWN 50
4.4.4 A CLAMPED MULTIPLE RADII CIRCULAR ARCH
SUBJECTED TO A VERTICAL CONCENTRATED LOAD
AT CROWN 50
4.4.5 A HINGED SEMI-CIRCULAR ARCH SUBJECTED TO
A VERTICAL CONCENTRATED LOAD AT CROWN 51
4.4.6 A CLAMPED SEMI-CIRCULAR ARCH SUBJECTED TO
A VERTICAL CONCENTRATED LOAD AT CROWN 51
4.4.7 ARCHES WITH DIFFERENT PROFILES 52
4.4.7.1 SEMI-ELLIPTIC ARCH 53
4.4.7.2 CIRCULAR ARCH 53
4.4.7.3 PARABOLIC ARCH 53
4.4.7.4 SINUSOIDAL ARCH 54
4.5 STRESSES AND AMPLIFICATION FACTORS 54

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CHAPTER Page

V. DISCUSSION AND CONCLUSION 57

5.1 DISCUSSION 57
5.1.1 COMPARISON WITH PREVIOUS WORKS 57
5.1.2 APPROACHES OF NONLINEAR ELASTIC ANALYSIS 59
5.1.3 NATURE OF [nl] AND [n2] MATRICES 60
5.2 SUMMARY AND CONCLUSION 66

TABLES 68

FIGURES 73

LIST OF REFERENCES 97

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APPENDICES
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A. NEWTON-RAPHSON METHOD FOR UPDATED COORDINATES 101

A.1 GENERAL 101

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A.2 INITIAL STRAIN STIFFNESS MATRIX, [ k£ ] 102

A.3 UPDATED COORDINATES PROCEDURE ° 107

B. INCREMENTAL STIFFNESS MATRICES, [nl] AND [n2], BASED ON

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THE AVERAGE AXIAL STRAIN MODEL 109

B.l THE FIRST ORDER INCREMENTAL STIFFNESS MATRIX, [nl] 109

B.2 THE SECOND ORDER INCREMENTAL STIFFNESS MATRIX, [n2] 117

C. COMPUTER PROGRAM

C.l GENERAL 128

C.2 DESCRIPTION OF SUBROUTINES 128
C.3 VARIABLES USED IN THE COMPUTER PROGRAM 129
C.4 INPUT DATA ARRANGEMENT 134
C.5 COMPUTER PROGRAM "NANCURVE" 138

D. INCREMENTAL STIFFNESS MATRICES, [nl] AND [n2], BASED ON

THE QUARTIC AXIAL STRAIN MODEL 198

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 LIST OF TABLES

TABLE Page

2-1 ACCURACY OF THE GEOMETRIC REPRESENTATION

FOR PARABOLIC ARCH (RISE-9.6", SPAN-48") 68

4-1 LINEAR EQUILIBRIUM OF A SEMI-CIRCULAR ARCH SUBJECTED

TO A CONCENTRATED IN-PLANE LOAD AT CROWN 70

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4-2 LINEAR EQUILIBRIUM OF A PARABOLIC ARCH SUBJECTED TO
A CONCENTRATED IN-PLANE LOAD AT CROWN 71

4-3 LINEAR EQUILIBRIUM OF A SEMI-CIRCULAR ARCH SUBJECTED

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TO A CONCENTRATED LATERAL LOAD AT CROWN 72
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 LIST OF FIGURES

FIGURE Page

1-1 LOAD-DEFLECTION RELATION 73

2-1 BEAM ELEMENT (Curved In The x-z Plane) 74

2-2 CROSS-SECTION OF PRISMATIC MEMBER 74

2-3 COORDINATE SYSTEMS 75

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2-4 TYPICAL ELEMENT 75

2-5 TYPICAL ELEMENT AFTER TRANSFORMATION TO ELEMENT

COORDINATE SYSTEM 76
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3-1 NEWTON-RAPHSON ITERATION 77
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4-1 LINEAR EQUILIBRIUM OF A SEMI-CIRCULAR ARCH SUBJECTED

TO A CONCENTRATED IN-PLANE LOAD AT CROWN 78

4-2 LINEAR EQUILIBRIUM OF A PARABOLIC ARCH SUBJECTED TO

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A CONCENTRATED IN-PLANE LOAD AT CROWN 79

4-3 LINEAR EQUILIBRIUM OF A SEMI-CIRCULAR ARCH SUBJECTED

TO A CONCENTRATED LATERAL LOAD AT CROWN 80

4-4 A 90°-HINGED CIRCULAR ARCH SUBJECTED TO UNIFORM

RADIAL LOAD (IN-PLANE BEHAVIOR) 81

4-5 A HINGED PARABOLIC ARCH SUBJECTED TO UNIFORM LOAD

ON HORIZONTAL PROJECTION (IN-PLANE BEHAVIOR) 82

4-6 A 90°-HINGED CIRCULAR ARCH SUBJECTED TO UNIFORM

RADIAL LOAD (OUT-OF-PLANE BEHAVIOR) 83

4-7 A HINGED PARABOLIC ARCH SUBJECTED TO UNIFORM LOAD

ON HORIZONTAL PROJECTION (OUT-OF-PLANE BEHAVIOR) 84

4-8 A 28°-CLAMPED CIRCULAR ARCH SUBJECTED TO

A VERTICAL CONCENTRATED LOAD AT CROWN 85

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FIGURE Page

4-9 A 60°-CLAMPED CIRCULAR ARCH SUBJECTED TO A VERTICAL

CONCENTRATED LOAD AT CROWN 86

4-10 A 60°-CLAMPED CIRCULAR ARCH SUBJECTED TO A SKEW

CONCENTRATED LOAD AT CROWN 87

4-11 A CLAMPED MULTIPLE RADII CIRCULAR ARCH SUBJECTED

TO A VERTICAL CONCENTRATED LOAD AT CROWN 88

4-12 A HINGED SEMI-CIRCULAR ARCH SUBJECTED TO

A VERTICAL CONCENTRATED LOAD AT CROWN 89

4-13 A CLAMPED SEMI-CIRCULAR ARCH SUBJECTED TO

A VERTICAL CONCENTRATED LOAD AT CROWN 90

4-14 ARCH AND FRAME PROFILES 91

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4-15 SINUSOIDAL, PARABOLIC, CIRCULAR, SEMI- ELLIPTIC ARCHES,
TRIANGULAR FRAME, AND RECTANGULAR FRAME SUBJECTED TO
A CONCENTRATED LOAD AT THEIR CROWNS 92
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4-16 STRESSES AND AMPLIFICATION FACTORS AT THE QUARTER
POINT A 93
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4-17 AXIAL FORCE AT CROWN 94

4-18 STRESSES AND AMPLIFICATION FACTOR AT CROWN 95

4-19 STRESSES AND AMPLIFICATION FACTORS AT THE CROWN 96

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A-1 CONFIGURATION OF A TWO DIMENSIONAL CURVED BEAM

ELEMENT AT SUCCESSIVE LOAD INCREMENTS IN UPDATED
LAGRANGE FORMULATION 101a

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 CHAPTER I

INTRODUCTION

1.1 GENERAL

The concept of basing structural design on ultimate strength has

gained increasing acceptance in recent years. In general, the

computation of the ultimate strength of a structure would involve load-

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displacement relationships that are nonlinear. In other words,

nonlinear analysis of structure becomes necessary. In the past, such

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analysis was shunned by engineers because it usually implies a large

amount of computations (in addition to theoretical complexities).

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However, current developments in computers are making such analysis

increasingly affordable for engineering practice.

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Nonlinear behavior of structures may be due to geometric changes ,

which represent the effect of distortion of the structure on its

response, or to material properties such as a nonlinear stress -s train

relation.

In the present study a procedure for the computation of nonlinear

elastic response of curved beam members is presented. Only geometric

nonlinearity is considered. This study was originated from a search of

an efficient method of nonlinear elastic analysis of arches or curved

structures in two and three dimensional space.

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This chapter describes the objective and scope of the present

work, a literature review of related studies, and the notation used in

the subsequent analysis.

1.2 OBJECTIVE AND SCOPE

Many engineering structures have components that may be considered

as curved beams. Several examples are the ribs of arch bridges, arch

frames, horizontally curved highway bridges, the components of aircraft

frames, ship frames, and vessel frames.

Figure 1-1 illustrates a load-displacement curve of a general arch

structure (herein the terms "arch" and "curved beam" are used

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interchangeably) which can be obtained by the solution of the nonlinear

equilibrium equations of the system. The curve "OCD" is called the

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"fundamental path". The point (C) on the fundamental path at which the

load is a relative maximum is called a "limit point". Depending on the

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properties of the arch and loading, a point of "bifurcation" may occur

before the limit point (i.e., point A) or after the limit point (i.e. ,
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point A'). Immediately beyond the bifurcation point on the fundamental

path, the structure is unstable, so that the response could follow the

secondary path AB or A'B' . If the bifurcation point occurs before the

limit point, the buckling shape would be "antisymmetrical" (sidesway).

If the bifurcation point occurs after the limit point, C, the arch

would have buckled at C in a "symmetrical" mode (snap through)

(48,22)*.

* Numbers in parantheses refer to entries in the list of references.

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It should be noted that the "classical buckling" theory would

assume that, up to the point when buckling takes place , A", the

structure would maintain its original undeformed shape. In other words,

the prebuckling deformation is neglected. At buckling, it goes into an

adjacent equilibrium configuration, B", which would then be unspecified

in magnitude.

Considerable amount of work has been done (see literature review)

on the development of suitable finite element models for the analysis

of curved beams. Most of the previous works have dealt with their

linear or stability analyses in the plane of the structure . Past

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studies that had considered out-of-plane behavior have been limited to

buckling analysis (as an eigenvalue problem). Such an analysis

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represents a short cut procedure to obtaining the ultimate load based

on the assumption of no displacement, or a linear relation between

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displacement and load, prior to buckling. Its application is limited to

those situations where the displacement at the incipient buckling is

small. For more general cases, i.e., when the latter displacement is
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not small, it becomes necessary to solve the nonlinear equilibrium

problem and obtain the corresponding load-displacement curve from which

the ultimate load could be determined.

Nonlinear equilibrium analysis is, in general, difficult to

formulate and expensive to carry out the numerical solution. Past

studies that dealt with such analysis of curved structures have been

limited to behavior in the plane of the arch.

The objective of the present study is to develop a three

dimensional nonlinear curved beam finite element which is applicable to

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both linear and nonlinear analyses of arbitrary geometry in two and

three dimensional space.

The curved beam element developed herein represents an improvement

of the previous model presented by Wen and Lange (45) for buckling

analysis. The geometric representation and the displacement functions

have been simplified for more convenient application. However, through

the use of the "average axial strain", the new model is found to be

substantially more effective and accurate than the previous one. The

curved axis of the element is represented by a second order polynomial

in the curvilinear coordinates. In deriving the linear stiffness

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matrix, the displacement functions are approximated by cubic

polynomials. The incremental (or nonlinear) stiffness matrices are

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derived by interpolating the transverse and longitudinal displacements

respectively by cubic and linear polynomials. A Lagrangian description

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of the arch displacements is used.

The present study uses the "incremental stiffness matrices" method

outlined by Mallett and Marcal (24). The strain energy is written in

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terms of the displacement variables. Geometrically nonlinear effects

are considered by including both the linear and quadratic terms of the

displacements in the expression for the generalized strains.

Furthermore, following Wen and Rahimzadeh (47), the quadratic

(nonlinear) terms are averaged over the element length . By using these

functions the expression for the strain energy of an element is

derived. This expression consists of three parts the quadratic ,

cubic , and quartic terms . Differentiating these expressions twice

yields the linear stiffness matrix, [k], and the first and second order

incremental stiffness matrices, [nl] and [n2], of the element. The

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linear stiffness matrix need be evaluated numerically by Gauss

quadrature method. However, because of the use of lower order

interpolation functions for some displacement components, terms of the

nonlinear stiffness matrices can be and are evaluated in closed form.

Assuming that the system is elastic and conservative, the

equilibrium equation is obtained from the first variation of the

potential energy. This represents a set of nonlinear algebraic

equations. The equation governing the linear incremental behavior is

obtained from the second variation of the potential energy.

The nonlinear equilibrium problem is solved by the Newton-Raphson

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method for a series of load increments. Possible instability along the

solution path is also tested by checking the determinant of the tangent

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stiffness matrix of the structure at every load increment. In

implementing this method, the convergence check is based on the

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unbalanced force vector.

A computer program was prepared for the implementation of the

above described nonlinear equilibrium analysis. Numerical results were

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obtained involving arches with in-plane and out-of-plane behavior.

Various types of loading and support condition were considered. To

provide some insight into the effects of variations in the arch profile

on its nonlinear response, semi - elliptic, circular, parabolic, and

sinusoidal shapes having the same rise to span ratio were considered.

The influence of the number of elements on the accuracy of the results

was investigated. The amplification factors for stresses and

displacement were also studied.

The problems were classified into small, intermediate, and large

displacement categories. The small displacement problems are those in

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which the deflection is less than about 2% of the arch span.

Intermediate displacement problems denote those in which the deflection

is of the order of 2%-25% of the arch span. Beyond 25% the problems are

called large displacement ones.

Comparisons of numerical results with those of other methods

indicates that the method presented is very accurate and efficient. The

procedure is generally not sensitive to the load step size. For all of

the numerical problems considered, accurate load-displacement curve may

be obtained by using at most eight elements to represent the entire

arch. For symmetrical problems, only one half of the arch (four

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elements) need be considered. Many cases required only two elements.

The method works very well for small and intermediate displacement
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problems. Most common practical problems would fall into these

categories.
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For very large displacements, it may be necessary to use the so

called "updated Lagrangian coordinates" method of solution as described

in Ref. 47 for straight beam elements. Such a procedure for the curved
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element is outlined in Appendix A.

The nonlinear elastic behavior of structures are often discussed

in terms of displacements. The ratio of the displacements obtained from

a nonlinear analysis to that obtained from a linear analysis is called

a (displacement) "amplification factor". Because of its importance in

design application, a look at the maximum stress is taken in this study

also. It was found that the amplification factor for stress was always

smaller than that of the displacement.

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1.3 LITERATURE REVIEW

For a straight beam finite element, it is well known that a cubic

polynomial assumed for the transverse displacement and a linear one for

the longitudinal displacement yield accurate results. Such is not the

case for curved beams.

1.3.1 LINEAR EQUILIBRIUM

Dawe (16) has studied the use of higher order polynomials as shape

functions for curved beams. He pointed out that there was a great

improvement gained by increasing the order of the assumed displacement

components from cubic to quintic. Gallert and Laursen (18) have

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presented a mixed formulation of finite elements for arches of

arbitrary shape. They established the convergence proof for this

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method. Numerical results indicated that the convergence is rapid.

Mebane and Stricklin (25) have pointed out that rigid body motion could
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be considered to be implicitly included in the polynomial form of shape

functions as the number of elements used to represent the structure

increases. Ashwell (1) discussed a class of curved finite elements

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(circular) whose shape functions were derived from independent

polynomial expressions for the generalized strains rather than

displacements. It was shown that the convergence of the strain element

was independent of arch types (shallow, thin moderate, thick moderate,

thin deep, and thick deep) and the behavior was superior to other

models.

1.3.2 BUCKLING ANALYSIS

For in-plane buckling analysis (as eigenproblems), Austin and Ross

(2) have compared the solutions of the in-plane buckling of

symmetrically loaded arches between the classical buckling theory and

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the exact, nonlinear buckling analysis. They found that except for

buckling in the symmetric mode (snap - through), the buckling load

obtained with the classical theory was very close to the bifurcation

load obtained with the exact theory. The conclusion, however, was based

on loading types that resulted in relatively small prebuckling

deformations.

Ojalvo and Newmann (30) have reported a basic theoretical work on

the linear elastic stability of a curved beam in space. Ojalvo, Demuts,

and Tokarz (29) followed the preceding work to study the out-of-plane

buckling of a member curved in one plane. Tokarz and Sandhu (40)

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developed the linear differential equations and obtained solutions for

the lateral-torsional buckling of a parabolic arch subjected to a

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uniformly distributed load. Wen and Lange (45) developed a finite

element model for a beam initially curved in one plane but deformable
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in three dimensional space. Geometric nonlinearities have been included

in the analysis. Linear as well as nonlinear e i genp r ob 1 ems were

formulated by setting the structural incremental stiffness to zero and

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assuming that the displacement increases linearly with the applied

loads. Their curved beam element could be used to calculate the in-

plane or out-of-plane buckling loads of arbitrary arch geometry. The

curved axis of the element was represented by a fourth order

polynomial. The displacement functions in the three dimensional space

were each approximated by a cubic polynomial.

1.3.3 NONLINEAR EQUILIBRIUM ANALYSIS

Mallett and Marcal (24) presented the general relationships

between the strain energy, the total equilibrium and incremental

equilibrium equations in terms of the usual linear stiffness matrix and

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two nonlinear incremental stiffness matrices. Wen and Rahimzadeh (47)

presented a "Finite Element Average" model for a three dimensional

nonlinear straight beam element, where the nonlinear part of the axial

strain was averaged over the element length. Without such averaging,

the element, which was formulated based on an application of the Mallet

and Marcal work (24), would generally be excessively stiff. The model,

which was based on the Lagrangian coordinate system, worked very well

for nonlinear analysis of frames in two or three dimensional space.

Solution procedures based on fixed and updated coordinate systems were

presented.

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For nonlinear equilibrium analysis of arches, Huddleston (20)

studied the inplane behavior of two hinged circular arches with any
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rise to span ratio by formulating the problem as a two point boundary

value problem consisting of six nonlinear, first order differential

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equations and appropriate boundary conditions. The theory was exact in

the sense that no restriction were placed on the size of the

deflections. The problem was solved by a "shooting method" in which

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the boundary value problem was converted to an initial value problem,

and the Regula-falsi procedure. The formulation was limited to two

hinged circular arches subjected to a vertical concentrated load at the

crown.

Noor, Green, and Hartley (27) developed a curved beam finite

element using the "mixed formulation" for the geometrically nonlinear

analysis of deep arches. While the displacement formulation adopted in

most works, including this one, and most general computer programs for

structural engineering uses only (generalized) displacements at the

nodes as degrees of freedom, the mixed formulation employs both

 10

displacements and forces at nodes as degrees of freedom. The

formulation was based on a nonlinear deep arch theory with the effect

of transverse shear deformation included. A total Lagrangian

description of the arch deformation and Lagrangian interpolation

functions were used in the formulation. Newton-Raphson method was used

to solve the resulting nonlinear equations. Circular, parabolic, and

semi-elliptic arches were analysed to obtain their inplane nonlinear

responses. They concluded that their mixed model performance was

accurate and less sensitive to variations in the arch geometry than

that of the displacement model.

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Belytschko and Glaum (5) presented a higher order corotational

formulation for the "initially curved beam element" (the shape of a

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bent beam which was straight before bending) in two dimension. The

displacement fields of each element are decomposed into rigid body and
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deformation displacements. The deformation displacements in the axial

and transverse directions are respectively described by linear and

cubic shape functions. The nonlinear equilibrium equation was solved by

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the modified Newton-Raphson method. The model was used to solve several

shallow arch problems. It was concluded that the higher order

corotational formulation converges to the exact solution more rapidly

than the lower order one.

Stolarski and Belytschko (37) pointed out that the preceding

curved beam element has the tendency to be too stiff unless the inplane

displacement field is of sufficiently high order polynomials. This

phenomenon was called "membrane locking". To eliminate this effect, the

reduced integration method (i.e., the numerical integration is

performed by using only one or two Gauss points) was used and shown to