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Nonlinear gap and Mindlin shell elements for the analysis of concrete structures

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Nonlinear gap and Mindlin shell elements for the analysis of concrete structures
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Nonlinear gap and Mindlin shell elements for the analysis of concrete structures
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Ahn, Kookjoon,
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NONLINEAR GAP AND MINDLIN SHELL ELEMENTS
FOR THE ANALYSIS OF CONCRETE STRUCTURES


















By

KOOKJOON AHN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


U::i: L "At; 7 OF FLORIN LIfRPMR17


1990














ACKNOWLEDGEMENTS


I would like to express my deep gratitude to professor

Marc I. Hoit for his invaluable guidance and support. I also

thank professors Clifford 0. Hays, Jr., Fernando E. Fagundo,

John M. Lybas, and Paul W. Chun for being on my committee. I

also express my gratitude to professor Duane S. Ellifritt

for his help as my academic advisor at the start of my Ph.D.

program.

Thanks are also due to all the other professors not

mentioned above and my fellow graduate students, Alfredo,

Jose, Mohan, Prasan, Prasit, Shiv, Tom, Vinax, and Yuh-yi.

Finally, I am thankful to every member of my family,

especially my wife and son, for their patience and support

in one way or another.

The work presented in this dissertation was partially

sponsored by the Florida Department of Transportation.















TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS ...................................... ii

ABSTRACT ...................................... ........ v

CHAPTERS

1 INTRODUCTION ....................... ............. 1

1.1 General Remarks ............................ 1
1.2 Link Element ............................... 2
1.3 Shell Element .............................. 5
1.4 Literature Review .......................... 5

2 GENERAL THEORIES OF NONLINEAR ANALYSIS .......... 13

2.1 Introduction ............................... 13
2.2 Motion of a Continuum ...................... 14
2.3 Principle of Virtual Work .................. 16
2.4 Updated Lagrangian Formulation ............. 18
2.5 Total Lagrangian Formulation ............... 22
2.6 Linearization of Equilibrium Equation ...... 26
2.7 Strain-Displacement relationship
Using von Karman Assumptions ............... 28

3 THREE-DIMENSIONAL LINK ELEMENT .................. 34

3.1 Element Description ...................... 34
3.2 Formation of Element Stiffness ............. 43
3.3 Solution Strategy .......................... 51
3.4 Element Verification ....................... 52

4 LINEAR SHELL ELEMENT ............................ 59

4.1 Introduction ............................... 59
4.2 Formulation of Shape Functions ............. 59
4.3 The Inverse of Jacobian Matrix ............. 64
4.4 Membrane Element ........................... 66
4.5 Plate Bending Element .................... 73

5 NONLINEAR SHELL ELEMENT ........................ 92

5.1 Introduction ............. ................. 92
5.2 Element Formulation ........................ 93


iii











5.3 Finite Element Discretization .............. 100
5.4 Derivation of Element Stiffness ............ 113
5.5 Calculation of Element Stiffness Matrix .... 115
5.6 Element Stress Recovery .................... 119
5.7 Internal Resisting Force Recovery ......... 122

6 NONLINEAR SHELL ELEMENT PERFORMANCE ............ 126

6.1 Introduction ............................... 126
6.2 Large Rotation of a Cantilever ............ 126
6.3 Square Plates .............................. 133

7 CONCLUSIONS AND RECOMMENDATIONS.................. 143

APPENDICES

A IMPLEMENTATION OF LINK ELEMENT .................. 146

B IMPLEMENTATION OF LINEAR SHELL ELEMENT ......... 170

C IMPLEMENTATION OF NONLINEAR SHELL ELEMENT ....... 219

REFERENCES ............................................ 230

SUPPLEMENTAL BIBLIOGRAPHY ............................ 238

BIOGRAPHICAL SKETCH ................................... 240














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONLINEAR GAP AND MINDLIN SHELL ELEMENTS
FOR THE ANALYSIS OF CONCRETE STRUCTURES

By

KOOKJOON AHN

August, 1990

Chairman: Marc I. Hoit
Major Department: Civil Engineering

Segmental post-tensioned concrete box girders with

shear keys have been used for medium to long span bridge

structures due to ease of fabrication and shorter duration

construction.

Current design methods are predominantly based on

linear elastic analysis with empirical constitutive laws

which do not properly quantify the nonlinear effects, and

are likely to provide a distorted view of the factor of

safety.

Two finite elements have been developed that render a

rational analysis of a structural system. The link element

is a two-dimensional friction gap element. It allows opening

and closing between the faces of the element, controlled by

the normal forces. The Mindlin flat shell element is a

combination of membrane element and Mindlin plate element.

This element considers the shear responses along the element









thickness direction. The shell element is used to model the

segment itself.

The link element is used to model dry joints and has

shown realistic element behavior. It opens under tension and

closes under compression. The link element has shown some

convergence problems and exhibited a cyclic behavior.

The linear Mindlin shell element to model the concrete

section of the hollow girder showed an excellent response

within its small displacement assumption.

The nonlinear Mindlin flat shell element has been

developed from the linear element to predict large

displacement and initial stress (geometric) nonlinearities.

The total Lagrangian formulation was used for the

description of motion. The incremental-iterative solution

strategy was used. It showed satisfactory results within the

limitation of moderate rotation.

Three areas of further studies are recommended. The

first is the special treatment of finite rotation which is

not a tensorial quantity. The second is the displacement

dependent loadings commonly used for shell elements. The

third is the material nonlinearity of concrete which is

essential to provide realistic structural response for safe

and cost effective designs.














CHAPTER 1
INTRODUCTION



1.1 General Remarks



In the past few decades segmental post-tensioned

concrete box girders have been used for medium to long span

bridge structures. Highway aesthetics through long spans,

economy due to ease of fabrication, shorter construction

duration are some of the many advantages of precast segment

bridge construction.

The segments are hollow box sections, match cast with

shear keys in a casting yard, then assembled in place,

leaving the joints entirely dry. The shear keys are meant to

transfer service level shears and to help in alignment

during erection.

Current design methods are heavily based on linear

elastic analysis with empirically derived constitutive laws

assuming homogeneous, isotropic materials. The behavior

under load of the bridge system is very complex. Analyses

which do not properly quantify the nonlinear effects

including the opening of joints in flexure, are likely to

provide a distorted view of the factor of safety existing in

a structural system between service loads and failure. The










potential sliding and separation at the joints due to shear,

and by deformations generated by temperature gradients over

the depth and width of the box further complicate the

problem [1].

Two finite elements have been developed that render a

rational analysis of the system. The link element is a two-

dimensional friction gap element. It allows sliding between

the faces of the element, controlled by a friction

coefficient and the normal forces. It also accounts for zero

stiffness in tension and a very high stiffness under

compression. This link element was borrowed from rock

mechanics and newly applied to this problem to model the dry

joint between the segments. The Mindlin flat shell element

is a combination of membrane element and Mindlin plate

element. This element considers the shear responses along

the element thickness direction. The shell element was used

to model the segment itself. This element can handle large

displacement and geometric nonlinearities.



1.2 Link Element



A link element is a nonlinear friction gap element used

to model discontinuous behavior in solid mechanics. Some

examples are interfaces between dissimilar materials and

joints, fractures in the material, and planes of weakness.

These have been modeled using constraint equations, discrete










springs and a quasi-continuum of small thickness [2]. The

following characteristics of prototype joints were

considered.


1. Joints can be represented as flat planes.

2. They offer high resistance to compression in the
normal direction but may deform somewhat modeling
compressible filling material or crushable
irregularities.

3. They have essentially no resistance to a net tension
force in the normal direction.

4. The shear strength of joints is frictional. Small
shear displacements probably occur as shear stress
builds up below the yield shear stress.


A model for the mechanics of jointed rocks was

developed by Goodman [3]. The finite element approximation

was done as a decomposition of the total potential energy of

a body into the sum of potential energies of all component

bodies. Therefore, element stiffness is derived in terms of

energy.

The Goodman element was tested for several modeled

cases.


1. Sliding of a joint with a tooth.

2. Intersection of joints.

3. Tunnel in a system of staggered blocks.


A problem with the Goodman's two dimensional model is

that adjacent elements can penetrate into each other.

Zienkiewicz et al. [4] advocate the use of continuous










isoparametric elements with a simple nonlinear material

property for shear and normal stresses, assuming uniform

strain in the thickness direction. Numerical difficulties

may arise from ill conditioning of the stiffness matrix due

to very large off-diagonal terms or very small diagonal

terms which are generated by these elements in certain

cases. A discrete finite element for joints was introduced

which avoids such theoretical difficulties and yet is able

to represent a wide range of joint properties, including

positive and negative dilatency (expansion and compaction

accompanying shear) [3].

The element uses relative displacements as the

independent degrees of freedom. The displacement degrees of

freedom of one side of the slip surface are transformed into

the relative displacements between the two sides of the slip

surface. This element has been incorporated into a general

finite element computer program [5]. The use of relative

displacement as an independent degree of freedom to avoid

numerical sensitivity is discussed in detail [6]. An

isoparametric formulation is given by Beer [2]. A four-node,

two-dimensional link element and a eight-node plate bending

element were used to model the dry jointed concrete box

girder bridge with shear keys [7].










1.3 Shell Element



The shell element is formulated through the combination

of two different elements, the membrane element and the

Mindlin plate bending element.

The Mindlin plate element is different from the

Kirchhoff plate element in that the former allows transverse

shear deformation while the latter does not.

The nonlinearities included in the formulation of the

flat shell element is for large displacement and geometric

nonlinearity due to initial stress effects. The large

displacement effects are caused by finite transverse

displacements. These effects are taken into account by

coupling transverse displacement and membrane displacements.

The initial stress effects are caused by the actual stresses

at the start of each iteration. These stresses change the

element stiffness for the subsequent iteration. These

effects are evaluated directly from the stresses at the

start of each iteration and included in the element

stiffness.



1.4 Literature Review



The purpose of nonlinear analysis is to develop the

capability for determining the nonlinear load-deflection

behavior of the structures up to failure so that a proper










evaluation of structural safety can be assured. There are

two general approaches for nonlinear analysis. The first

approach is a linearized incremental formulation by reducing

the analysis to a sequence of linear solutions. The second

approach is mathematical iterative techniques applied to the

governing nonlinear equations [8].

The advantage of the incremental approach results from

the simplicity and generality of the incremental equations

written in matrix form. Such equations are readily

programmed in general form for computer solutions [9].

A generalized incremental equilibrium equation for

nonlinear analysis can be found in [10, 11, 12]. The

formulation is valid for both geometrical and material

nonlinearities, large displacements and rotations,

conservative and displacement dependent (nonconservative)

loads.

There are two frames for the description of motion. The

difference lies in the coordinate systems in which the

motion is described. These are the total Lagrangian

formulation which refers to the initial configuration [10,

11] and the updated Lagrangian formulation which refers to

the deformed configuration [12]. There have evolved two

types of notations in the description of motion. A

correlation is given these two notations, the B-notations

and the N-notations, currently used in the Lagrangian

formulation of geometrically nonlinear analysis [13]. A










short history of early theoretical development of nonlinear

analysis can be found in [9, 14].

One form of updated Lagrangian formulation is the

corotational stretch theory [15].

Shell elements are often derived from governing

equations based on a classical shell theory. Starting from

the field equations of the three-dimensional theory, various

assumptions lead to a shell theory. This reduction from

three to two dimensions is combined with an analytical

integration over the thickness and is in many cases

performed on arbitrary geometry. Static and kinematic

resultants are used. These are referred to as classical

shell elements. Alternatively, one can obtain shell elements

by modifying a continuum element to comply with shell

assumptions without resorting to a shell theory. These are

known as degenerated shell elements. This approach was

originally introduced by Ahmad, Irons, and Zienciewicz [16,

17]. Other applications can be found in [8, 18-25].

In large rotation analysis, the major problems arise

from the verification of the kinematic assumptions. The

displacement representation contains the unknown rotations

of the normal in the arguments of trigonometric functions.

Thus additional nonlinearity occurs. Further difficulties

enter through the incremental procedure. Rotations are not

tensorial variables, therefore, they cannot be summed up in

an arbitrary manner [17]. One of the special treatment of










finite rotation is that the rotation of the coordinate

system is assumed to be accomplished by two successive

rotations, an out-of-plane rotation followed by an in-plane

rotation using updated Lagrangian formulation [26, 27].

Usually the loadings are assumed to be conservative,

i.e., they are assumed not to change as the structure

deforms. One of the well known exceptions is pressure

loading which can be classified as conservative loading or a

nonconservative loading [28]. Another is the concentrated

loading that follows the deformed structure. For example, a

tip loading on a cantilever beam will change its direction

as the deformation gets larger. As loading is a vector

quantity, the change in direction means that the loading is

not conservative. Sometimes this is called a follower

loading.

The governing equation for large strain analysis can be

used for small increments of strain and large increments of

rotations [29]. This can be regarded as a generalization of

nonlinearity of small strain with large displacement. If

large strain nonlinearity is employed, an important question

is which constitutive equation should be used [9].

The degree of continuity of finite element refers to

the order of partial differential of displacements with

respect to its coordinate system. Order zero means

displacement itself must be continuous over the connected

elements. Order one means that the first order differential










of displacement must be continuous. Thus the higher order

the continuity requirement, the higher the order of assumed

displacement (shape, interpolation) function.

Mindlin-Reissner elements require only Co continuity,

so that much lower order shape functions can be used,

whereas in Kirchhoff-Love type elements, high order shape

functions must be used to satisfy the C1 continuity.

Furthermore, since Mindlin-Reissner elements account for

transverse shear, these elements can be used for a much

larger range of shell thickness. The relaxed continuity

requirements which permit the use of isoparametric mapping

techniques gives good computational efficiency if formulated

in the form of resultant stresses [30]. Unlike compressible

continuum elements, which are quite insensitive to the order

of the quadrature rule, curved Co shell elements require

very precisely designed integration scheme. Too many

integration points result in locking phenomena, while using

an insufficient number of quadrature points results in rank

deficiency or spurious modes [30]. While Gauss point stress

results are very accurate for shallow and deep, regular and

distorted meshes, the nodal stresses of the quadratic

isoparametric Mindlin shell element are in great error

because of the reduced integration scheme which is necessary

to avoid locking [31].

The degenerate solid shell element based on the

conventional assumed displacement method suffers from the










locking effect as shell thickness becomes small due to the

condition of zero inplane strain and zero transverse shear

strain. Element free of locking for linear shell analysis

using the formulation based on the Hellinger-Reissner

principle with independent strain as variables in addition

to displacement is presented in [32].

Shear locking is the locking phenomenon associated with

the development of spurious transverse shear strain.

Membrane locking is the locking phenomenon associated with

the development of non-zero membrane strain under a state of

constant curvature. Machine locking is the locking

phenomenon associated with the different order of dependence

of the flexural and real transverse shear strain energies on

the element thickness ratio, and it is therefore strictly

related to the machine finite word length [33]. Some of the

solutions are as follows:


1. Assumed strain stabilization procedure using the Hu-
Washizu or Hellinger-Reissner variational principles
[33].

2. The assumed strain or mixed interpolation approach [34,
35].

3. Suppressing shear with assumed stress/strain field in a
hybrid/mixed formulation [30]. Suppression of zero
energy deformation mode using assumed stress finite
element [36].

4. Coupled use of reduced integration and nonconforming
modes in quadratic Mindlin plate element [37].

5. Higher order shallow shell element, with 17 to 25 nodes
[38, 39].









6. Global spurious mode filtering [40].

7. Artificial stiffening of element to eliminating zero
energy mode, special stabilizing element [41].

In the faceted elements, due to the faceted

approximation of the shell surface, coupling between the

membrane and the flexural actions is excluded within each

individual element, the coupling is, however, achieved in

the global model through the local to global coordinate

transformation for the elements [39].

In geometrically nonlinear analysis with flat plate

elements, it is common to use the von Karman assumptions

when evaluating the strain-displacement relations. The

assumption invoked is that the derivatives of the inplane

displacements can be considered to be small and hence their

quadratic variations neglected. However, this simplification

of the nonlinear strain-displacement relationship of the

plate, when used in conjunction with the total Lagrangian

approach, implies that the resulting formulation is valid

only when the rotation of the element from its initial

configuration is moderate. Thus for the total Lagrangian

approach to handle large rotations, simplifications of the

kinematic relationship using the von Karman assumptions is

not permitted [39].

Some of the special solution strategies to pass the

limit point are given in references [25, 42-48]. A limit

point is characterized by the magnitude of tangential









stiffness. It is zero or infinite at a limit point. Thus

conventional solution strategies fail at the limit point.

Arc length method was introduced in reference [42], and

applied in the case of cracking of concrete [43]. This was

improved with line search and accelerations in references

[44, 45]. Line search means the calculation of an optimum

scalar step length parameter which scales the standard

iterative vector. This can be applied to load and

displacement control and arc length methods [44].

The traditional solution strategies are iterative

solutions, for example, Newton-Raphson, constant stiffness,

initial stiffness, constant displacement iteration, load

increment [46] along with Cholesky algorithm with shifts for

the eigensolution of symmetric matrices [47] for element

testing for spurious displacement mode.

The vector iteration method without forming tangent

stiffness for the postbuckling analysis of spatial

structures is also noted [48].

The linearized incremental formulation in total

Lagrangian description has been used for this study of large

displacement nonlinearity including initial stress effects.

The special treatment of finite rotation is not included in

the current study. Material nonlinearity is also excluded.













CHAPTER 2
GENERAL THEORIES OF NONLINEAR ANALYSIS



2.1 Introduction



The incremental formulations of motion in this chapter

closely follow the paper by Bathe, Ramm, and Wilson [11].

Other references are also available [9, 10, 12, 14, 15, 49,

50, 51].

Using the principle of virtual work, the incremental

finite element formulations for nonlinear analysis can be

derived. Time steps are used as load steps for static

nonlinear analysis. The general formulations include large

displacements, large strains and material nonlinearities.

Basically, two different approaches have been pursued

in incremental nonlinear finite element analysis. In the

first, Updated Lagrangian Formulation, static and kinematic

variables, i.e., forces, stresses, displacements, and

strains, are referred to an updated deformed configuration

in each load step. In the second, Total Lagrangian

Formulation, static and kinematic variables are referred to

the initial undeformed configuration.

It is noted that using either of two formulations

should give the same results because they are based on the










same continuum mechanics principles including all nonlinear

effects. Therefore, the question of which formulation should

be used merely depends on the relative numerical

effectiveness of the methods.



2.2 Motion of a Continuum



Consider the motion of a body in a Cartesian coordinate

system as shown in Fig. 2-1. The body assumes the

equilibrium positions at the discrete time points 0, dt,

2dt, ..., where dt is an increment in time. Assume that the

solution for the static and kinematic variables for all time

steps from time 0 to time t, inclusive, have been solved,

and that the solution for time t+dt is required next.

The superscript on left hand side of a variable shows

the time at which the variable is measured, while the

subscript on left hand side of a variable indicates the

reference configuration to which the variable is measured.

Thus the coordinates describing the configuration of the

body using index notation are


At time 0 = x

At time t = txi

At time t+dt = t+dtx














P t+dt
P( x )


t
P( Xi)


0
P( Xi)


Fig. 2-1 Motion of a Body









The total displacements of the body are

At time 0 = ui

At time t = tui

At time t+dt = t+dtu

The configurations are denoted as

At time 0 = C

At time t = tc

At time t+dt = t+dtc

Thus, the updated coordinates at time t and time t+dt are
txi = Oxi + tui

t+dtx = Oxi + t+dtu

The unknown incremental displacements from time t to

time t+dt are denoted as (Note that there is no superscript

at left hand side.)

u = t+dtui tui (2.1)



2.3 Principle of Virtual Work


Since the solution for the configuration at time t+dt

is required, the principle of virtual work is applied to the

equilibrium configuration at time t+dt. This means all the

variables are those at time t+dt and are measured in the

configuration at time t+dt and all the integration are

performed over the area or volume in the configuration at

time t+dt. Then the internal virtual work (IVW) by the









corresponding virtual strain due to virtual displacement in
t+dtC is


S t+dt t+dt t+dt
t+dt ij t+dt eij (+dt dV) (2.2)


where,
t+dt rj = Stresses at time t+dt measured in the
t+dt j configuration at time t+dt.

= Cauchy stresses.

= True stresses.
t+dt
t+dt eij = Cauchy's infinitesimal(linear) strain tensor
referred to the configuration at time t+dt.

= Virtual strain tensor.

6 = Delta operator for variation.

and the external virtual work (EVW) by surface tractions and

body forces is


EVW = t+dt t ] 6 [ t+dt u (t+dt dA)
t+dt k t+dt k (t+dt



St+dt t+dt b 6 t+dt u (t+dt dV)
St+dt t+dt k t+dt uk

(2.3)
where,
t+dt tk = Surface traction at time t+dt measured in
t+dt the configuration at time t+dt.

t+dt Uk = Total displacement at time t+dt measured in
t+dt the configuration at time t+dt.









6 t+dt Uk = Variation in total displacement at time
t+dt t+dt measured in configuration at time t+dt

= Virtual displacement.
t+dt p = Mass density per unit volume.
t+dt

t+dt bk = Body force per unit mass.
t+dt k


and all the integration is performed over the area and the

volume at time t+dt.


2.4 Updated Lagrangian Formulation


In this formulation all the variables in Eqs. (2.2) and

(2.3) are referred to the updated configuration of the body,

i.e, the configuration at time t. The equilibrium position

at time t+dt is sought for the unknown incremental

displacements from time t to t+dt.

The internal virtual work, the volume integral in Eq.

(2.2) measured in the configuration at time t+dt can be

transformed to the volume integral measured in the

configuration at time t in a similar manner that is given in

reference [52]


IVW t+dt t+dt e (t+dt dV)
t+dt ij t+dt i (t dV)



t Sij 6 Et ij (t dV) = EVW (2.4)
I ~t t~d










where,
t+dt Sj = Second Piola-Kirchhoff (PK-II) stresses
t measured in the configuration at time t.

6 t+dt ej = Variations in Green-Lagrange (GL) strain
t tensor measured in the configuration at
time t.


The PK-II stress tensor at time t+dt, measured in the

configuration at time t can be decomposed as

t+dt t t
Sij =t Sij + t Sij j + t Sij (2.5)



because the second PK-II stress at time t measured in the

configuration at time t is the Cauchy stress.

From Eq. (2.1), the total displacements at time t+dt

measured in the configuration at time t is

t+dt t
t Ui = t Ui + Ui = t ui (2.6)


This is true because the displacement at time t measured in

the configuration at time t is zero. In other words, the

displacement at time t+dt with respect to the configuration

at time t is the incremental displacement itself.

And the GL strain is defined in terms of displacement as


Eij = 1 (Ui,j + Uji + Uk,iUk,j) (2.7)

E and U are used in the places of e and u to avoid confusion

between general strain and incremental strain, and between









general displacement and incremental displacement used in
this formulation. It is noted that these finite strain

components involve only linear and quadratic terms in the

components of the displacement gradient. This is the

complete finite strain tensor and not a second order

approximation to it. Thus this is completely general for any

three-dimensional continuum [52].

Then the GL strain tensor at time t+dt measured at time

t can be calculated as


+dt = i [(ui+ i)+ u) + ( j + tuj),i

+{(uk + tuk),i)( tuk + uk),j)]


= tuij + tuj,i+ tuki tuk,j]


= teij + tij

= tij (2.8)

where,

ttij = tei + t ij

= Incremental GL strain in tC.

tej = (ui,j + tuji )

= Linear portion of incremental GL strain in t.

= This is linear in terms of unknown incremental
displacement.
= Linearized incremental GL strain in tc.










ttij = t (uk,i tuk,j)

= Nonlinear portion of incremental GL strain.

The variations in Green-Lagrange strain tensor at time
t+dt measured in the configuration at time t can be shown as
using Eq. (2.8).

t+dt t
6 t ij = 6 ( tij + Ei ) = 6 t (2.9)


6teij = 0 because teij is known. There is no variation in
known quantity.

Then using the Eqs. (2.5), (2.8) and (2.9), the

integrand of Eq. (2.4) becomes

t+dt t+dt t
t Sij 6 qt =ij ( t i + tSij ) 6 tCij

= trij + tSij)(6 teij + 6tij)

=tij(6eij + 6t 1ij) + t~ij 6 te + rij 6 tij
t t
=tij 6 ti ij + ti 6eJ 6 (2.10)
tj t t ttij 6 t tii j` t


The constitutive relation between incremental PK-II

stresses and GL strains are


tSij = tCijkj t'kl


(2.11)








Finally the equilibrium Eq. (2.4) from the principle of
virtual work using Eqs. (2.10) and (2.11) is


I tCijkl tkl 6 tcij tdv

+ tri 6 t Vij tdV


= EVW J ij 6 eij tdV
t 1t
(2.12)
where, the external virtual work must be transformed from
t+dtc to tC. This is not applicable to conservative loading,

i.e., loading that is not changed during deformation.


EVW= t+dt tk [ t+dt uk (tdA)


+ t+dt t+dt b t+dt td) (2 )



and this is the general nonlinear incremental equilibrium
equation of updated Lagrangian formulation.


2.5 Total Lacrangian Formulation


Total Lagrangian formulation is almost identical with
the updated Lagrangian formulation. All the static and
kinematic variables in Eqs. (2.2) and (2.3) are referred to









the initial undeformed configuration of the body, i.e, the

configuration at time 0. The terms in the linearized strain

are also slightly different from those of updated Lagrangian

formulation.

The volume integral in Eq. (2.2) measured in the

configuration at time t+dt can be transformed to the volume

integral measured in the configuration at time 0 as [52]


t+dt 'i 6 t+dt eij t+dt
t+dt ]i t+dt


o t Sij t+dt j (dV) (2.14)



where,
t+dtS = Second Piola-Kirchhoff stress tensor
0 measured in the configuration at time 0.

6 t+dtij = Variations in Green-Lagrange (GL) strain
0 tensor measured in the configuration at
time 0.

The PK-II stress tensor at time t+dt, measured in the

configuration at time 0 can be decomposed as

t+dtsij= tsij+ *ij (2.15)
o1 ) o) 1 o01j (2.15)

From Eq. (2.1), the total displacements at time t+dt

measured in the configuration at time 0 is










t+dtu = u + Ui (2.16)
o 0i o0 o


Then the GL strain tensor at time t+dt measured at time
0 can be calculated as
t+dt = [(tui + oui)j + (tuj + ou i



=tt+
+(( e*k+ oUk),i)(( Uk+ Uk) j)]

= Eij + e + i

= ij + Eij (2.17)

where,
tt tutu tu tu)
oij = I (t u + + )
0 o o rJ o k,i k,j

= GL strain at time t in oC.

ij = oeij + oij

= Incremental GL strain in oC.

oeij = (oi, + oUj1 + uk,i ouk,kj+ k,j ouk,i)
= Linear portion of incremental GL strain in oC.
=This is linear in terms of unknown incremental
displacement.
= Linearized incremental GL strain in oC.

oij = i (oUk, ouk,j)

= Nonlinear portion of incremental GL strain.

The variations in Green-Lagrange strain tensor at time
t+dt measured in the configuration at time t can be shown as









using Eq. (2.17).

Stdtij = 6 ( ij+ ij ) = 6 (2.18)
o i= oec olu olt


6 te, = 0 because ej is known. There is no variation in
o ij o-J
known quantity.

Then using the Eqs. (2.15), (2.17) and (2.18), the

integrand of Eq. (2.14) becomes

t+dt t+dt ts
t+dt*i 6 t+dti ( tSi + S ) 6 e
o o o o0J o'J

= (ij oSij) oeij + 6 oiij)

= Sij(6 eij + 6 ij) + oSij 6 e 6 +

= Si 6 eij + tS i 6 ei + ij 6 o7ij (2.19)


The constitutive relation between incremental PK-II

stresses and GL strains are


Sij = oCijkj ockl (2.20)


Finally the equilibrium Eq. (2.14) from the principle


of virtual work using Eqs. (2.19) and (2.20) is











I oCijkl oekl 6 j dV


+ tsi 6 oij OdV


= EVW tij 6 e dV
o 1 oe ij
(2.21)

where, the external virtual work must be transformed from
t+dtc to oC. This is not applicable to conservative loading,

that is, loading that is not changed during deformation.


EVW = t+dt t t+dt uk (dA)



+ t+dt p t+dt b t+dt uk (odV) (2.22)



and this is the general nonlinear incremental equilibrium

equation of total Lagrangian formulation.


2.6 Linearization of Equilibrium Equation


The incremental strain from time t to t+dt is assumed

to be linear, i.e., ekl = ekl in Eqs. (2.11), (2.12),

(2.20), and (2,21).









For the updated Lagrangian formulation,


tSij = Cijkj tekl (2.23)

and,


ItCijkl tekl teij d


tI t
+ tij 6 tij tdV


= EVW rj 6 e tdV
(2.24)

For the total Lagrangian formulation,


oSij = oCijkj oekl (2.25)

and,


IoCijkl oekl 6 oeij odV
ddV


+ [sij ij dV


= EVW Sj e odV
o 0J o ij
(2.26)
It should be noted that the surface tractions and the
body forces in the calculation of external virtual work may
be treated configuration dependent when the structure
undergoes large displacements or large strains. If this is









the case, the external forces must be transformed to the

current configuration at each iteration [10, 11, 12].



2.7 Strain-Displacement Relationship
Using the von Karman Assumptions


The nonlinear strain terms can be simplified for the

plate or shell type structures using von Karman assumption

of large rotation.

In the mechanics of continuum the measure of

deformation is represented by the strain tensor Eij [52] and

is given by using index notation.


2Eij = ( ui,j + uj,i + k,iuk,j ) (2.27)

where,

ui = Displacement in i-direction.

uij = aui / axj

xi = Rectangular Cartesian coordinate axes, i=1,2,3.

uk,iuk,j = ul,iul,j + u2,iu2,j + u3,iu3,j

The von Karman theory of plate is a nonlinear theory

that allows for comparatively large rotations of line

elements originally normal to the middle surface of plate.

This plate theory assumes that the strains and rotations are

both small compared to unity, so that we can ignore the

changes in geometry in the definition of stress components

and in the limits of integration needed for work and energy









considerations [53]. It is also assumed that the order of

the strains is much less than the order of rotations.

If the linear strain eij and the linear rotation rij

are defined as

2eij = uij + uji (2.28)

2rij = u ui (2.29)

Then the sum of Eqs. (2.28) and (2.29) gives

2(eij + rij) = 2uij (2.30)

and the subtraction of Eq. (2.29) from Eq. (2.28) gives

2(eij rij) = 2uj (2.31)

From Eqs. (2.30) and (2.31), it is concluded that


uk,j = ekj + rkj (2.32)

uk,i = eik rik (2.33)

Eq. (2.33) can be rewritten as


uk,i = eki + rki (2.34)

since eik = eki from the symmetry of linear strain terms and

rik = -rki from the skew symmetry of the linear rotation
terms.

The strain-displacement Eq. (2.27) now becomes


2Eij = 2eij + (eki + rki)(ekj + rkj)


(2.35)









by substituting Eqs. (2.30) through (2.34) into Eq. (2.27).

Thus the nonlinear strain terms have been decomposed into

linear strain terms and linear rotation terms.

From the assumption on the order of strains and

rotations


eki << rki and ekj << rkj (2.36)

Thus Eq. (2.35) can be simplified as by ignoring eki and

ekj.

2Eij = 2eij + rkirkj (2.37)


The straight line remains normal to the middle surface

and unextended in the Kirchhoff assumption, but it is not

necessarily normal to the middle surface for the Mindlin

assumption. For both assumptions the generic displacements

u,v,w can be expressed by the displacements at middle

surface.

For the Kirchhoff plate [20],


u(x,y,z) = uo(x,y) z[Wo(x,y),x]

v(x,y,z) = vo(x,y) z[wo(x,y),y] (2.38)

w(x,y,z) = Wo(x,y)
where,

uo, Vo, wo = Displacements of the middle surface
in the direction of x, y, z.

u, v, w = Displacements of an arbitrary point
in the direction of x, y, z.









Now the linear strain components eij and the linear rotation
components rij can be calculated using Eqs. (2.28) and
(2.29).

ell = (ul,1 + Ul,1) = ul,1 = ,x
el2 = (l,2 + u2,1) = 1(uly + Vx)

el3 = (u1,3 + U3,1) = (-Wox + Wx)
22 = i(u2,2 + u2,2) = u2,2 = Vy (2.39)
e23 = (u2,3 + u3,2) = (-woy + Wy)

e33 = (u3,3 + u3,3) = u3,3 =

The rotation terms r12, r13, r23 are the rotation quantities
about the axes 3(z), 2(y) and l(x), respectively. For the
plate located in the xy plane, the rotation about z axis rl2
is much smaller than rotation about x axis r23 and y axis
r13 and therefore rl2 is assumed to be zero here. And it is
noted further that wo(x,y) is the same as w(x,y) and is a
function of only x and y so that w,3 = w,z = 0.


lrl21 << lr231 or 1r13 (2.40)

rl = (ul,1 ul,1) = 0

r12 = 1(ul,2 u2,1) = i(u, Vx) = 0

r13 = I(ul,3 u3,1) = (-Wox Wx) = -Wx
r22 = 1(u2,2 u2,2) = 0 (2.41)

23 = 1(u2,3 u3,2) = (-Wo'y W'y) = -Wy
r33 = i(u3,3 3,3) = 0









The linear strain component eij is symmetric and the linear

rotation component rij is antisymmetric.


eij = eji
rij = -rji (2.42)


The strain components from Eq. (2.37) can be rewritten using

Eqs. (2.39) and (2.41).


Ex = ell + (r112 +

yy = e22 + (r122 +
Ezz = e3 + 1(r132 +

Exy = el2 + 2(rllrl2
Exz = e3 + (rllrl3

Eyz = e23 + U(rl2r13


r21 + r31 ) = ell + r31

r22 + r322) = e22 + r32

r232 + r332) = 1(r132 + r232) = 0

+ r21r22 + r31r32) = el2 + r31r32
+ r21r23 + r31r33)

+ r22r23 + r32r33) (2.43)


Egz term is assumed to be zero because it does not have the

linear term. Exz and Eyz terms are transverse shear terms

which can be ignored for thin plate. Then Eq. (2.43) can be

rearranged as follows using Eqs. (2.41) if all the zero

terms are removed.


Exx = ell + r312 = ell + (W,x)2
Eyy = e22 + r322 = 22 + )2

Ey = el2 + Ir31r32 = e2 + (W'x) (,y)

Exz = e13

Eyz = e23


(2.44)







33


Thus the decomposition of exact strain components has been

done using the Kirchhoff plate assumptions (2.38) and the

von Karman assumption (2.40) on the magnitude of rotation.

It is noted that all the inplane displacement gradients in

nonlinear strain terms are ignored through von Karman

assumptions [20]. This fact will be applied in chapter 5.













CHAPTER 3
THREE DIMENSIONAL LINK ELEMENT



3.1 Element Description



The link element used here is based on the two

dimensional element developed by Cleary [54]. The link

element is based on the following assumptions.

Any normal compressive force is transferred to the

other side of the link without any loss. To facilitate this,

a very limited amount of loss through displacement should be

allowed. Currently, this limited displacement is defaulted

to .001 units, while it is a input parameter. The link

separates in response to any net tension, losing its normal

stiffness.

To discuss the shear force transfer, some definitions

for friction are needed. The force to start one body sliding

along the other body is called the static friction force.

The force to keep it moving is the kinetic friction force.

There are two corresponding coefficients of friction, static

friction coefficient and dynamic friction coefficient, where

the static friction coefficient will generally be greater

than the dynamic friction coefficient.









Two laws of friction were used in the link element. The

first law is that the frictional force is proportional to

the normal force, with the constant of proportionality being

the friction coefficient. The second law is that friction

does not depend on the apparent area of the connecting

solids, i.e., it is independent of the size of the bodies.

The shear force is transferred through friction. The

uncertainty in friction is the factor which limits the

overall accuracy of the calculation. Therefore, it is

assumed that the static friction coefficient is proportional

to the dynamic friction coefficient. For nonmetallic

materials, the ratio of dynamic coefficient to static

coefficient is about 0.75.

The link element is composed of two surfaces. If the

shear force is less than or equal to the static friction

force, i.e., coefficient of friction times the normal force,

the shear force is balanced by the friction force and the

total force is transferred. This is shown in Fig. 3-1. But

if the shear force is greater than the static friction

force, one surface of the link element will move along the

other surface. In this case there will be a dynamic friction

force which is less than the shear force. This dynamic

friction force can only resist a portion of the shear and

the system is not in static equilibrium. Therefore, if the

shear force is greater than the static friction force, the

link element will lose its shear stiffness. This can also be










modeled with a body on roller and spring as shown in Fig. 3-

2. The spring model of the link element is shown in Fig. 3-

3.

The link element here has four nodes and each node has

three translational degrees of freedom in local u-, n-, and

w-directions. The total number of element degrees of freedom

is 12. The element degrees of freedom are shown in Fig. 3-4.

The equivalent "strain" for the link element is defined

as the average deformation at the center of the element. The

average deformation corresponding to the translational

degrees of freedom, i.e., uo, vo, and wo, can be directly

calculated from the joint displacements by averaging the

difference in nodal displacements at the ends of element in

local u-, n-, and w-direction in turn. The relative rotation

at the center of the element, ro, can be found using nodal

displacements in local n-direction and the element length as

shown in Fig. 3-5. This angle is not an "average" value but

the "relative" change in angle of the center line due to

rotation.

The two joint parameters must be introduced. These are

kn, the unit stiffness normal to the joint, and ks, the unit

stiffness along the joint.

The off-diagonal term kns to account for dilatation

during shearing is ignored because this joint element will

model the dry joint between concrete box girder bridge

segments. No significant dilation is expected in this case.










Some values of kn and ks were reported in geotechnique

area [3]. As the values are those for natural joints, they

do not directly apply to this case.

From the test results [7], it can be seen that the

shear stiffness of dry joint ranges from 70,000 to 286,000

psi per inch at ultimate. In case of frictional strength,

this can be interpreted as linear behavior between the

origin and the ultimate point.

It seems reasonable that the normal stiffness of the

element, kn, is assumed to be stiffer than the connected

material by the order of 103 to transfer the normal force

without any significant loss. The forces are either totally

transferred in compression or totally lost in tension. The

latter case has no problem related to the value of kn.

The shear stiffness parameter is more difficult to

define. The data available is so limited that even a

statistical treatment cannot be done. But in the analysis of

structural behavior up to the ultimate, these properties do

not have great influence because the forces are transferred

through friction.

The shear stiffness becomes zero upon sliding. But

there is some 'residual' shear force. This 'residual' force

is equal to friction force. Therefore, if shear displacement

is more than the displacement just before the sliding the

shear stiffness is set to be zero.

























Ff


P, N = External forces.

F = Friction force.
f


m = Friction coefficient






1) P < or = mN then P = Friction Force. In Equilibrium.

2) P > mN then the body moves but the frictional force

mN is acting against the other body.


Fig. 3-1 Friction Force







FRICTIONAL SPRING WITH SHEAR STIFFNESS
Force in spring = mN
N


-%AAN-


BEFORE


SLIDING


SFRICTIONAL SPRING WITH ZERO STIFFNESS


N


A F

mN


mN < F

AFTER SLIDING


Fig. 3-2 Spring Model of Friction Force


__ __






















Kn = Zero under tension.
= Very large under
compression.


SPRING MODEL FOR NORMAL FORCES


SPRING MODEL FOR SHEAR FORCES


Fig. 3-3 Spring Model of Link Element











11


K


Fig. 3-4 Element Degrees of Freedom of Link Element


L

12

I

34






CENTER OF ELEMENT
K


UK
NoLNo


uo = [(UK + UL) (UI + UJ)] / 2

ro= p a

S[(VK-VL) (VJ-VI)] / L


VK


O------



VI
o o-


Fig. 3-5 Element "Strain"


uo-_ZL









3.2 Formation of Element Stiffness


There are four nodes per element. Each node has three

degrees of freedom corresponding the translational

displacements in u-, n-, and w-direction resulting in 12

element degrees of freedom as shown in Fig. 3-4. The element

stiffness is derived directly from the physical behavior of

the element described in section 3.1.

The mathematical symbol {} is used for a column vector

and [] for a matrix. The nodal displacement column vector

{q)(12) is composed of 12 translational nodal displacements

corresponding to the 12 element degrees of freedom.


(q} = { ui vi wi Uj Vj Wj uk vk wk ul vl wl )T

The "strain" is defined as the average deformation at

the center of the element as shown in Fig. 3-5. The "strain"

column vector {e)(4) is


{e} = { uo vo wo ro )T


where,

uo = ( uk + ul ) / 2 ( u + j ) / 2

o = vk + 1 ) / 2 ( + Vj ) / 2

w = (wk + 1 ) / 2 ( i + j ) / 2

ro = ( vk v ) / L ( vj vi ) / L

where,

L = The length of the element.











uo, vo, wo = Average nodal displacements in local
u-, n-, w-directions.

ro = The relative angle change about local z axis.


Therefore the relationship between "strain" and nodal

displacements is


(e}(4) = [B](4,12) (q)(12)


The [B](4,12) matrix which gives strains due to unit

values of nodal displacements is



-0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0
0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0
0.0 0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5
0.0 1/L 0.0 0.0 -1/L 0.0 0.0 1/L 0.0 0.0 -1/L 0.0



The "stress" is defined as the normal and shear stress

per unit of area. {s} is the average stress on the surface

due to the two nodal forces exerted in the plane of the

surface. This stress is in equilibrium with the stress on

the other surface of the element as shown in Fig. 3-6. m is

the moment of the nodal forces on one surface in local n-

direction about the center of the element. This moment is

also balanced by the moment of the nodal forces on the other

surface of the element. This moment is used to define the

distribution of the normal stress of the element as shown in

Fig. 3-7.










Pnl

fl1


P...'


uk


-~- -~-
-~- -~- -~-
-~- -~- -~-
-~- -~-


uj -nj



Pj J ~pnj


-W- Stresses

-- Nodal Forces


Local Coordinate System


Fig. 3-6 Nodal Forces and Stresses of Link Element


/


ni


-W.- -.- -w

-0--o 0--0 p











FORCE TRANSFER THROUGH ONE EDGE
OF THE LINK ELEMENT






P



L K
Vo ro

2 Vo


CENTER OF ELEMENT


p J


Fig. 3-7 Element "Strain", m









The "stress" column vector (s}(4) is


(s) = { Sx, sn, Sz, m )

The "stress-strain" relationship is


(s)(4) = [E](4,4) {e)(4)

where,



kx 0 0 0
[E] = 0 kn 0 0
0 0 kz 0
0 0 0 km



where km can be related to kn using the definition of

the moment m, i.e.,

m = (Sn)(L)(t)(0.5)(L)

= kmro = (km)(vo/(0.5)(L))



Thus, km = (knVo)(L)(t)(0.5)(L) / [Vo/(0.5)(L)]

= (0.25)(t)(kn)(L3)


where, t = Element thickness.

This assumes that there is no coupling between the

shear stress and normal stress.

The element nodal force column matrix (P)(12) is

composed of the 12 nodal forces shown in Fig. 3-6.










(P) Pui Pni Pwi Puj Pnj Pwj Puk Pnk Pwk

Pul nl Pwl T

Stress can then be related to nodal forces using the

definition of stress and force equilibrium between the two

surfaces of the element.

By the definition of stress,


sn = (1/Lt)( Pnk + Pnl ) (3.1)

sx = (1/Lt)( Puk + Pul ) (3.2)

sz = (1/Lt)( Pwk + Pwl ) (3.3)
m = Pnk(0.5)(L) Pnl(0.5)(L) (3.4)
where, Lt = (L)(t)

By force equilibrium of the two surfaces,

Pi = -P and Pj = Pk (3.5)

To express the element nodal forces in terms of the
stress, we use Eqs. (3.1) through (3.5) to find the force

recovery matrix [FR]. [FR] gives the nodal forces in

equilibrium with the element stresses.

From (Eq. (3.1) + Eq. (3.4)),

2Pnk = (L)(t) (sn) + 2(m)/L

Pnk = 0.5(L) (t) (sn) + (l/L)(m)









From Eq. (3.5),

Pj = -Pk
nj = -Pnk
= -0.5(L)(t)(sn) (1/L)(m)

From Eq. (3.1),


Pn1 = (L)(t)(sn) Pnk
= (L)(t)(sn) ( 0.5(L)(t)(sn) + (1/L)(m))
= 0.5(L) (t) (n) (1/L) (m)

From Eq. (3.5),

Pni = Pn1
= -0.5(L)(t)(sn) + (1/L)(m)

From the assumption that Puk = Pul and Eq. (3.2),


Puk = (L)(t)(sx)/2

Pul = (L)(t)(sx)/2

From Eq. (3.5),


Pui = ul = -(L)(t)(sx)/2

Puj = uk = -(L)(t)(sx)/2

From the assumption that Pwk = Pwl and Eq. (3.3),


Pwk = (L)(t)(sz)/2
Pwl = (L)(t)(sz)/2










From eqn 5,


Pwi = wl = -(L)(t)(sz)/2

Pwj = Pwk = -(L)(t) (s)/2


Therefore, the force-stress relationship is


{P)(12) = [FR](12,4) (s)(4)


where the force recovery matrix [FR](12,4) is


[FR] =


-Lt/2
0
0
-Lt/2
0
0
Lt/2
0
0
Lt/2
0
0


0
-Lt/2
0
0
-Lt/2
0
0
Lt/2
0
0
Lt/2
0


0
0
-Lt/2
0
0
-Lt/2
0
0
Lt/2
0
0
Lt/2


0
1/L
0
0
-1/L
0
0
1/L
0
0
-1/L
0


And this relationship is further expanded using the

stress-strain relationship and the strain-nodal displacement

relationship as follows.


(P}(12) = [FR](12,4)

= [FR](12,4)

= [Bt](12,4)


[E](4,4) {e)(4)

[E](4,4) [B](4,12)

[E](4,4) [B](4,12)


Then finally this can be symbolized as equilibrium

equation.


(q)(12)

{q)(12)










{P)(12)= [Ke](12,12) (q}(12)


where [Ke] = [Bt][E][B]

Here it is noted that [FR] = [Bt] and [Ke] = [Bt][E][B] just

as in the case of common finite element method.

The final element stiffness matrix [Ke] is


(L)t

4


kx
0
0
kx
0
0
-kx
0
0
-kx
0
0


0
2kn
0
0
0
0
0
0
0
0
-2kn
0


0
0
kz
0
0
kz
0
0
-kz
0
0
-kz


kx
0
0
kx
0
0
-kx
0
o
-kx
0
0


0
0
0
0
2kn
0
0
-2kn
0
0
0
0


0
0
kz
0
0
kz
0
0
-kz
0
0
-kz


-kx
0
0
-kx
0
0
kx
0
0
kx
0
0


0
0
0
0
-2kn
0
0
2kn
0
0
0
0


0
0
-kz
0
0
-kz
0
0
kz
0
0
kz


-kx
0
0
-kx
0
0
kx
0
0
kx
0
0


0
-2kn
0
0
0
0
0
0
0
0
2kn
0


0
0
-kz
0
0
-kz
0
0
kz
0
0
kz


This matrix can be

standard rotation.


rotated to any direction using the


3.3 Solution Strategy


The structural stiffness changes because of the slip

and debonding of the link. Therefore, the process of the

resistance of the total structure physically becomes

nonlinear. Correspondingly, special solution techniques for

nonlinear behavior are needed.

This can be done using the iterative solution technique

with initial stiffness or tangent stiffness. The latter can










be formed by assembling the structural stiffness at the

beginning of each iteration and this converges faster than

the initial stiffness.

A third solution strategy for this case is event-to-

event technique which is usually employed for the linear

stiffnesses between any two "events," which are defined as

the intersection point between two linear segments. This

also provides means of controlling the equilibrium error.

Any significant event occurring within any element

determines a substep. The tangent stiffness is modified in

each substep, and hence, the solution closely follows the

exact response.

3.4 Element Verification



3.4.1 SIMPAL

The finite element analysis program SIMPAL [55], is

used to implement and verify the element formulation. SIMPAL

was chosen for the initial implementation because that was

the original implementation done by Cleary [54]. This way,

the 3-D aspects could be implemented and verified using

Cleary's original program. A table of the element

verification is shown in Fig. 3-8 and Fig. 3-9.









LOADING


10




10
Y
1 2


NODE THEORY SIMPAL ERROR

DISP 2 -.1333 -.1333 .000

DISP 4 -.1333 -.1333 .000

STRESS N/A -80 -80 .000


NODE THEORY SIMPAL ERROR

DISP 2 -.1333 -.1333 .000


DISP 4 -.1333 -.1333 .000

STRESS N/A -80 -80 .000



NODE THEORY SIMPAL ERROR


DISP 2 -.1333 -.1337 .003

DISP 4 -.1333 -.1337 .003

STRESS N/A -80 -80 .000


* NODE 2 Y DISP =-0.1017-04 Z DISP = -0.8684-05
NODE 4 Z DISP =-0.8684-05 Y DISP =-0.1017-04
2 2
SQRT((.1017) + (.08684) )=0.1337


Fig. 3-8 Link Element Test Using SIMPAL


3
3


THICKNESS = .25
Ks = 3E6
Kn = 6E6


RESULTS








LOADING


4 6
Aft Am


.P -......


3 5
1 _~~


RESULTS


8
Aft


Ii
MP, -


Adh -


10


1 I


1 3 5 7 Y
1 3 5 7 Y


Z
2


4 6


3 5


Fig. 3-9 Combined Test Model for SIMPAL


Z
2


-~araooa a;a8oo~ uru -


ia--m i-- i-o -n


----------


HtS~ ^~ k


10
8


10










3.4.2 ANSR



The test examples used are the same as those used in

the initial element verification using SIMPAL. The results

from ANSR [56] are exactly the same as those from SIMPAL.

The link element was tested further using a modeled membrane

element composed of 22 truss elements as a membrane element

was not available at the time of element verification in

ANSR. The results are shown in Table 3-1 and the structures

used are shown in Fig. 3-10 and Fig. 3-11.












Table 3-1 Displacements of
for ANSR


Truss Model


Node Truss Truss Diff.
No. only w/ LINK (%)

10-x -.1027e-4 -.1049e-4 2.2

10-y -.1990e-5 -.2010e-5 0.9

ll-x -.9017e-5 -.9211e-5 2.2

11-y -.4906e-6 -.4973e-6 1.4

12-x -.9915e-5 -.1049e-4 2.2

12-y +.9742e-6 +.9703e-6 0.4











LINK ELEMENT


20




.2

20







.2


20


Fig. 3-10 Combined Test Model for ANSR

















































Fig. 3-11 Truss Model for ANSR














CHAPTER 4
LINEAR SHELL ELEMENT


4.1 Element Description



The shell element is formulated through a combination

of two different elements, the membrane element and the

Mindlin plate bending element [57].

The Mindlin plate element is different from the

Kirchhoff plate element in that the former allows transverse

shear deformation while the latter does not.

The common portions of the formulation of two elements

are


1. Formation of the shape functions.

2. Formation of the inverse of Jacobian matrix.


These processes can be done at the same time. The four-

to nine-node shape functions and their derivatives in rs-

space can be formed and then transformed into xy-space

through the inverse of Jacobian matrix.



4.2 Formulation of Shape Functions



The formulation of shape functions starts with three

basic sets of shape functions shown in Fig. 4-1.










1. The bilinear shape functions for four-node element.

2. The linear-quadratic shape functions for nodes
five to eight of the eight-node element.

3. The bubble shape function for node nine of
nine-node element.


These shape functions can be formulated directly from

the local coordinates of the element nodes through the

multiplication of the equations of the lines which have zero

values in the assumed displacement shapes and the scale

factor to force the shape function value to one at the node

for which the shape function is formed. The derivative of

each shape function with respect to r and s is then

evaluated from the shape function expressed in terms of r

and s.

If node nine exists, the value at node nine of shape

functions one to eight must be set to zero. The value of the

bilinear shape functions for a four-node element at node

nine is one fourth and the value of the linear-quadratic

shape functions for the five- to eight-node element at the

node nine is one half. This can be forced to zero using the

bubble shape function of the nine-node element because this

shape function has the value of one at node nine and zero at

all other nodes. Therefore the modification is the

subtraction of one fourth of the value the bubble shape

function has at node nine from the each shape function for

the corner nodes and the subtraction of one half of the










value of the bubble shape function of node nine for the

nodes five to eight, whichever exists.

If any of the center nodes on the edge of the element

(any one of nodes five to eight) exists, the bilinear shape

functions of four-node element must be modified further

because the value at center of the edge is one half in those

bilinear shape functions. This can be done by subtracting

one half of the linear-quadratic shape function for the

newly defined center node on the edge of the element from

the bilinear shape functions of the two adjacent corner

nodes. The value of any five node shape functions at the

corner node is zero. Therefore, no further consideration is

needed except for the shifting of the shape functions in the

computer implementation. These processes are shown in Fig.

4-2.

If any of the linear-quadratic shape functions of nodes

five to eight is missing, all the linear-quadratic shape

functions thereafter and the bubble shape function must be

shifted to the proper shape function number. For example, if

linear-quadratic shape function five is missing, then the

shape functions six to eight must be shifted to five through

seven and the bubble shape function must be shifted to the

node eight because all of the linear-quadratic shape

functions have been defined and numbered as shape functions

for the nodes five through eight and the bubble shape

function for the node nine.


















Four Node Element



Shape Function for Corner Node


Five Node Element


Shape Function for Edge Center Node








Nine Node Element




Shape Function for Element Center Node


Fig. 4-1 Three Basic Shape Functions























SF 1


SF 2


SF 3


SF 3


SF 4 = (SF 1) (1/4) (SF 3)


SF 5 = (SF 2) (1/2) (SF 3)


Fig. 4-2 Formation of Shape Functions









4.3 The Inverse of Jacobian Matrix



While the generic displacements are expressed in terms

of rs-coordinate, the partial differential with respect to

the xy-coordinate is needed for the calculation of strain

components. Thus the inverse of the Jacobian matrix must be

calculated. This can directly be found from the chain rule

using the notation (a,b) defined as the partial differential

of function a with respect to the variable b for simplicity.


f,x = (f,r)(r,x) + (f,s)(s,x)

f,y = (f,r)(r,y) + (f,s)(s,y)


In matrix form,



f,x r,x s,x f,r Jll-1 J12-1 f,r

f,y r,y s,y f,s J21-1 J22-1 f,s



The inverse of Jacobian matrix


But the terms in the inverse of the Jacobian matrix are not

readily available because the rs-coordinate cannot be solved

explicitly in terms of xy-coordinate. On the other hand, for

the isoparametric formulation, the geometry is interpolated

using the nodal coordinate values(constants) and the

displacement shape functions in terms of r and s. Thus the

generic coordinate x and y can be expressed in r and s









easily and explicit partial differentials of x and y with

respect to r and s can be performed. Therefore the Jacobian

matrix is computed and then inverted.

The Jacobian matrix is derived by the chain rule.


f,r = (f,x)( x,r) + (f,y)( y,r)

f,s = (f,x)( x,s) + (f,y)( y,s)


In matrix form,



f,r x,r y,r f,x Jll J12 f,x

f,s x,s y,s f,y J21 J22 f,



Jacobian matrix


nn
Let s be .
i=1

where nn = number of nodes (4 to 9).

From geometric interpolation equations,


x = Z fi*xi

y = Z fi*Yi


The terms in the Jacobian matrix are


J11 = x,r = (z fi*xi),r = Z ((fi,r) xi)

J12 = y,r = (Z fi*yi),r = Z ((fi,r) yi)

J21 = x,s = (Z fi*xi),s = Z ((fi,s) xi)

J22 = y,s = (Z fi*yi),s = Z ((fi,s) yi)











xi, yi are coordinate values of the element and are

constants and therefore can be taken out of the partial

differentiation.

The inverse of two-by-two Jacobian matrix can be found

-i
Jl-1 = r,x = J22 / det(J)
-i
J12 = s,x = -J12 / det(J)
-I
J21- = r,y = -J21 / det(J)
-i
J22- = s,y = J11 / det(J)

where det(J) = J11J22 12J21



4.4 Membrane Element



The formulation of the membrane element used for the

implementation follows the procedure shown on pages 115

through 118 in reference [57]. The ( ) symbol will be used

for the column vectors.

Nodal displacements are the nodal values of two in-

plane translations and denoted as {ui vi}T. The generic

displacements are defined as two translational displacements

at a point and denoted as { u v )T. By the word generic it

is meant that the displacement is measured at an arbitrary

point within an element. The generic displacements u and v

can be calculated using shape functions. The shape function

is a continuous, smooth function defined over the closed










element domain and is differentiable over the open domain of

the element. The shape function is also the contribution of

displacement of a node for which the shape function has been

defined to the generic displacement. Thus the generic

displacement at an arbitrary point can be found by summing

up all the contributions of all the nodes of the element.

The displacement interpolation equations are


u = Z fi Ui

v = Z fi Vi


In the isoparametric formulation the geometry is

interpolated using the same shape function assumed for the

displacement interpolation.

Therefore, the geometry interpolation is


x = Z fi xi

y = Z fi Yi


where,

fi = Shape function for node i.

xi, Yi = Coordinates of node i.

ui, vi = Displacements at the node i.

u, v = Displacements at an arbitrary point within

an element.









The three in-plane strain components for a membrane

element are


{ E ) = ( ex Cy 7xy )

These strain components can be found through the

partial differentials of the generic displacements with

respect to xy-coordinates.

ex = ux

Ey = v,y

7xy = uy + VX

Using the inverse of the Jacobian matrix, the strain

components can be evaluated.


eX

= u,x

= (u,r)(r,x) + (u,s)(s,x)

= (u,r)(J11-1) + (u,s)(J12-1)

= ((Efiui),r)(J11-1) + ((EfiUi),s)(J12-1)

= Z[(fi,r)(r,x) + (fi,s)(s,x)] ui



y
= v,y
= (v,r)(r,y) + (v,s)(s,y)

= (v,r)(J21-1) + (v,s)(J22-1

= ((.fivi),r)(J21-1) + ((Zfivi),s)(J22-1)

= [(fi,r)(r,y) + (fi,s)(s,y)] vi











7xy
= u,y + v,x

= [(u,r)(r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)]

= [((Zfiui),r)(J21-1) + ((Zfiui),s)(J22-1)]

+ [((Zfivi),r)(J11-) + ((Zfivi),s)(J12-1)]
= Z[(fi,r)(r,y) + (fi,s)(s,y)] ui

+ Z[(fi,r)(r,x) + (fi,s)(s,x)] vi


New notations are introduced here to simplify the

equations. These are ai and bi and defined as follows:


ai = (r,x)(fi,r) + (s,x)(fi,s) = fi,x

bi = (r,y)(fi,r) + (s,y)(fi,s) = fiY

Then the strain terms above become


ex = Zaiui = Zf,x ui

Ey = Zbivi = Zfiy vi

7xy = Zbiui + Zaivi = Zfi,y ui + Zfi,x vi
In matrix form,



ex ai 0 ui
yE = 0 bi vi
7xy bi ai


In symbolic form,


[E] = E[Bi]Cqi]









where,



ai 0 fix 0
[Bi] = 0 bi 0 fi',
bi ai fi'y fix


and,


ui
[q] = Vi
vi


Therefore the strain at an arbitrary point within an

element is


[e] = [Bl][q1] + [B2][q2] + ... + [B9][q99


= [ B1 B2 B3 B4 B5 B6 B7 B8 B9 ]


ql
q2
q3
q4
95
q6
q7
q8
q9


The size of the vectors and matrix are


[e(3,1)] = [B(3,18)][q(18,1)]


In the actual calculation, this can be done by summing

up the [Bij[qi] over all the nodes for the given coordinates

of the point under consideration, i.e., the coordinates of

one of the integration points.










The stresses corresponding to the strains are

{ a } = { ax ay rxy )T

The stress-strain relationship of an isotropic material

is



Ell E12 0
[E] = E21 E22 0
0 0 E33


where,

E11 = E22 = E / ( 1 2 )

E12 = E21 = pE / ( 1 p2)

E33 = G

where,

E = Young's modulus

p = Poisson's ratio

G = shear mQdulus = E / ( 2*(1+p))

The integration over volume must be introduced for the

calculation of the element stiffness and equivalent nodal

loads of distributed loads or temperature effects. As the

thickness of the element is constant, the integration over

volume can be changed to the integration over area.

The element stiffness related to the degrees of freedom

of the node i can be calculated through the volume


integration of Bi(2,3)E(3,3)Bi(3,2).










[Ki] = BT E Bi dV


As [B] and [E] are constant about z, the integration

through the thickness from -t/2 to t/2 can be performed on z

only and yields


T
[Ki] = [ Bi(t)EBi] dA
A B

T -
= [ Bi E Bi] dA
A


where,


E = tE


The size of membrane element stiffness is 18 by 18.


[K] =


[ E ] [ B1 B2 B3 ... B8 B9 ] dV
(3,3)
(3,18)


B8
B9

(18,3)


Equivalent nodal loads due to body forces on the

membrane element are calculated as


Pb Jv fTbdV =11 1-1


fTbIJI dr ds


in which {b} = ( 0 0 bz T or { 0 b 0 }T or { bx 0 0 } in










accordance with the direction of gravity in the coordinate

system used. The nonzero quantities bx, by, or bz represent

the body force per unit area in the direction of

application.

Equivalent loads caused by initial(temperature) strains

are



PO = BTEEO dV
JV
1 1
BTEcIJI dr ds
S-1 J-i

where,

(C0)= exxO yyO 0 0 0 )T
= {aAT aAT 0 0 0 )



4.5 Plate Bending Element



The formulation of the plate bending element used for

the implementation has followed the procedures shown on

pages 217 through 221 in reference [57]. The { } symbol will

be used for the column vectors.

Many plate bending elements have been proposed. The

most commonly used are Kirchhoff plate elements and Mindlin

plate elements.










Kirchhoff theory is applicable to thin plates, in which

transverse shear deformation is neglected. The assumptions

made on the displacement field are


1. All the points on the midplane(z = 0) deform only
in the thickness direction as the plate deforms in
bending. Thus there is no stretching of midplane.

2. A material line that is straight and normal to the
midplane before loading is to remain straight and
normal to the midplane after loading. Thus there is
no transverse shear deformation (change in angle
from the normal angle).

3. All the points not on the midplane have displacement
components u and v only in the x and y direction,
respectively. Thus there is no thickness change
through the deformation.


Strain energy in the Kirchhoff plate is determined entirely

by in-plane strains ex, Cy, and 7xy which can be determined

by the displacement field w(x,y) in the thickness direction.

The interelement continuity of boundary-normal slopes is not

preserved through any form of constraint.

Mindlin theory considers bending deformation and

transverse shear deformation. Therefore, this theory can be

used to analyze thick plates as well as thin plates. When

this theory is used for thin plates, however, they may be

less accurate than Kirchhoff theory because of transverse

shear deformation. The assumptions made on the displacement

field are


1. A material line that is straight and normal to the
midplane before loading is to remain straight but
not necessarily normal to the midplane after









loading. Thus transverse shear deformation (change
in angle from normal angle) is allowed.

2. The motion of a point on the midplane is not
governed by the slopes (w,x) and (w,y) as in
Kirchhoff theory. Rather its motion depends on
rotations Ox and 0 of the lines that were normal to
the midplane of the undeformed plate. Thus 0, and 0
are independent of the lateral displacement w, i.e.,
they are not equal to (w,x) or (w,y).


It is noted that if the thin plate limit is approached, -xz

= 7yz = 0 because there is no transverse shear deformation.
In this case the angles 0x and 0y can be equated to the

(w,x) and (w,y) numerically but the second assumption still

holds.

The stiffness matrix of a Mindlin plate element is

composed of a bending stiffness [kb] and a transverse shear

stiffness [ks]. [kb] is associated with in-plane strains ex,

cy, and 7xy. [ks] is associated with transverse shear

strains 7xz and 7yz. As these two groups of strains are

uncoupled, i.e., one group of the strains do not produce the

other group of strains, the element stiffness can be shown

as [82]



[k] = ( BbEBb ) dA + (BsEBs) dA


because BbEBs = BsEBb = 0 from uncoupling (corresponding E =

0). Each integration point used for the calculation of [kg]

places two constraints to a Mindlin plate element,

associated with two transverse shear strains lyz and yzx. If










too many integration points are used, there will be too many

constraints in transverse shear terms, resulting in locking.

Therefore, a reduced or selective integration can prevent

shear locking. Or, the transverse shear deformation can be

redefined to avoid such locking.

For example, a bilinear Mindlin plate element responds

properly to pure bending with either reduced or selective

integration. But full two-by-two integration is used for

pure bending, shear strains appear at the Gauss points as

shown in Fig 4-3. As the element becomes thin, its stiffness

is due almost entirely to transverse shear. Thus, if fully

integrated, a bilinear Mindlin plate element exhibits almost

no bending deformations, i.e., the mesh "locks" against

bending deformations.

Nodal displacements for the plate bending consist of

one out-of-plane translation and two out-of-plane rotations

and are denoted as { wi 0xi 8yi )T. The rotations are chosen

independently of the transverse displacement and are not

related to it by differentiation. Thus the transverse shear

strains jxz and 7yz are considered in the formulation

resulting in five strain components. The generic

displacements are defined as three translational

displacements and denoted as { u v w }T. By the word generic

it is meant that the displacement is measured at an

arbitrary point within an element. These generic

displacements are different quantities from the nodal










displacements and therefore must be related to the nodal

displacements.

The generic displacements u and v can be calculated as

functions of the generic out-of-plane rotations using the

small strain(rotation) assumption. The relationship between

generic displacements and rotation is shown in Fig 4-4.


u = zBy

v = -ZBx


The generic displacements Ox and By can be found using the

assumed displacement shape functions and the corresponding

nodal displacements Oxi and 0yi.

The generic displacement w does not need any conversion

because it corresponds to the nodal displacement wi.

In the isoparametric formulation the geometry is

interpolated using the same shape function assumed for the

displacement interpolation.

The displacement interpolation is


8x = Z fi 0xi

6y = Z fi 0yi

w = Z fi Wi


Similarly, the geometric interpolation is


x = Z fi xi

y = Z fi Yi


















Zero Shear
Strain


One Point Gauss Integration


I Non-zero
Shear Strain


Two Point Gauss Integration


Fig. 4-3 Shear Strains at Gauss Point(s)


i~Amlh









Z

+ u



y

X



Positive small rotational angle about y-axis gives

positive generic displacement in x-direction ( u ).

Shown is xz-plane.


Positive small rotational angle about x-axis gives
negative generic displacement in y-direction ( v
Shown is yz-plane.


Fig. 4-4 Displacements due to Rotations










where,

fi = shape function for node i

xi, Yi = coordinates of node i



Therefore,


u = Zoy = z z fi 6yi

v = -zOx = -z Z fi Oxi

W = E fi Wi


The five strain components for plate bending element

are { ex ey 7xy 7xz Iyz )T. These strain components can be

found through the partial differentials of the generic

displacements with respect to xy-coordinates.


e = uI,

Ey = v,y

7xy = uy + V,X

7XZ = u,z + W,X

7yz = vz + w,y

Using the inverse of the Jacobian matrix found, the

strain components can be evaluated.


Ex

= u,X = (ZBy),X

= (u,r)(r,x) + (u,s)(s,x)

= (u,r)(J11-1) + (u,s)(J12-1)









= ((ZzfiOyi),r)(J~l-1) + ((zzfiyi) ,) (J12-1)
= z Z[(fi,r)(r,x) + (fi,s)(s,x)] Oyi




= v,y = (-zOx),y
= (v,r)(r,y) + (v,s)(s,y)
= (v,r)(J21-1) + (v,s)(J22-1)
= ((-zzfioxi),r)(J21-1) + ((-zZfi0xi),s)(J22-1)
= -z Z[(fi,r)(r,y) + (fi,s)(s,y)] Oxi


7xy
= u,y + v,x = (zoy),y + (-zox),X
= [(u,r)(r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)]
= [((zzfioyi),r)(J21-1) + ((zZfiyi),s)(J22-1)]
+ [((-zZfisxi),r)(Jll-1) + ((-zzfixi),s) (J12-1)]
= z Z[(fi,r)(r,y) + (fi,s)(s,y)] Oyi
-z Z[(fi,r)(r,x) + (fi,s)(s,x)] exi


7xz = (u,z) + (w,x)
= (ZOy),Z + w,x = 9y + w,x

= Zfisyi + [(w,r)(r,x) + (w,s)(s,x)]
= Zfioyi + [((Zfiwi),r)(J11-1) + ((Zfiwi),s) (J12-1)
= Zfieyi + [((Zfi,r)wi) (J11-1) + ((Zfi,s)wi)(J12-1)]
= ZfiOyi + z[(fi,r)(r,x) + (fi,s)(s,x)] wi










lyz = (v,z) + (w,y)
= (-z0x),z + w,y = (-OX) + w,y

= (-Zfioxi) + [(w,r)(r,y) + (w,s)(s,y)]

= (-Zfioxi) + [((Zfiwi),r)(J21-1) +((Zfiwi),s)(J22-1)]

= (-Zfioxi) + [((Zfi,r)wi)(J21-1) +((Zfi,s)wi)(J22-1)]

= (-Zfi0xi) + Z[(fi,r)(r,y) +(fi,s)(s,y)] Wi


New notations are introduced here to simplify the

equations. These are ai and bi and defined as follows:


ai = (r,x)(fi,r) + (s,x)(fi,s) =

bi = (r,y)(fi,r) + (s,y)(fi,s) =

Then the strain terms above become


f.l
i1,x


ex = z aioyi

Cy = -z Zbixi

7xy = z Zbiyi z Zaixi

7xz = fi0yi + Zaiwi

Tyz = fioxi + Zbiwi


In matrix form,



eX
fy
7xy =
7XZ
Lyz

In symbolic form,


[e] = E[Bi][qi]


0
-zbi
-zai
0
-fi
i


zai
0
zbi
fi
0


wi
exi
Oyi










where,


[Bi] =




or,




[Bi] =


0
- zbi
- zai
0
fi
i


0
0
0
fi,x
fi'y


zai
0
zbi
fi
0


0
- zfi,y
- zfi,x
0
fi


zfi,x
0
zfi,Y
fi
0


and,


[qi]


wi
ixi
Oyi


Therefore the strain at an arbitrary point within an

element is


[e] = [B][ql] ] + ... + [B2[ + + 9][q9]


= [ B1 B2 B3 B4 B5 B6 B7 Bg B9 ]


q91
92
93
94
95
96
97
98
99










The size of the vectors and matrix are


[E(5,1)] = [B(5,27)][q(27,1)]


In the actual calculation, this can be done by summing

up the [Bi][qi] over all the nodes for the given coordinates

of the point under consideration, i.e., the coordinates of

one integration point.

The stresses corresponding to the strains are


{ o ) = ( ax ay rxy rxz yz )TT

The stress-strain relationship of an isotropic material

is



E11 E12 0 0 0
E21 E22 0 0 0
[E] = 0 E33 0 0
0 0 0 E44 0
0 0 0 0 E55


where,

E11 = E22 = E / ( 1 p2 )

E12 = E21 = pE / ( 1 p2)

E33 = G

E44 = E55 = G / 1.2

where,

p = Poisson's ratio

G = shear modulus = E / ( 2*(1+p))









The form factor 1.2 for the E44 and E55 terms is
provided to account for the parabolic distribution of the
transverse shear stress rzx over a rectangular section. This
form factor 1.2 can be shown from the difference in
deflections of a cantilever beam at its free end [58].
Let a beam have a rectangular cross section of
dimensions b by t with a length of L. If P is the transverse
shear force, then the parabolic distribution of the
transverse shear stress rz, is

r'X = (3P/2bt3)(t2 4z2)
where z = 0 at the neutral axis.
Then the transverse shear strain energy from the parabolic
distribution can be calculated by


Us = (1/2)V (zx 2/G) dV


= (1/2) [((3P/2bt3)(t2 4z2))2 / G] dV


= [(1/2)(3P/2bt3)2]/G (t2 4z22 dAdz


= (area)[(1(/)(3P/2bt3)2]/G (t2 4z2)2 dz


= bL[(1/2)(3P/2bt3)2]/G (t2 4z22 dz

= 1.2(P2L/btG)/2









While the transverse shear strain energy from the constant

distribution is


Us = (1/2) I(rx2/G) dV


= (1/2) ((P/bt)2 /G) dV


= [(1/2)(P/bt)2/G] (btL)

= (p2L/btG)/2

This result suggests the view that a uniform stress

P/bt acts over a modified area bt/1.2, so that the same Us

results. Therefore the deflection at the free end of a

cantilever beam with parabolically distributed transverse

shear stress will be 1.2 times that with constantly

distributed transverse shear stress.

The Mindlin plate is generalized from the Mindlin beam.

Thus the higher transverse shear stiffness from the

assumption of constant transverse shear stress has been

reduced by dividing the corresponding elastic constants by

the factor 1.2 for flat plate element. The reduced stiffness

will produce the more flexible response in shear that is

expected from the actual parabolic distribution.

The integration over volume must be introduced for the

calculation of the element stiffness and equivalent nodal

loads of distributed loads or temperature effects. As the

thickness of the element is constant, the integration over









volume can be changed to the integration over area. This can

be accomplished through the partition of the strain-nodal

displacement matrix [Bi] as follows:



0 0 zfj,x
0 zfi,y 0
[Bi] = 0 zfi,x zfi,y

fi,x 0 fi
fi,Y fi 0


The submatrices are named as follows:



BiA ZBiA
[Bi] =
BiB BiB


The element stiffness can be calculated through the
T
volume integration of Bi (3,5)E(5,5)Bi(5,3). Thus the [E]

matrix is to be partitioned as follows:



Ell E12 0 0 0
E21 E22 0 0 0
[E] = 0 0 E33 0 0

0 0 0 E44 0
0 0 0 0 E55



The submatrices are named as follows:


E EA 0
E =
0 EB










Then the stiffness of the element is



FT B -T T [EA 0 ziA 1
[Ki] =I B E Bi dV = zBiA [ A B dV
0 EB BiB

2 -T T
= [ BA EA BA] + [BB EB BB]


As [B] and [E] are constant about z, the integration

through the thickness from -t/2 to t/2 can be performed on z

only and yields


= T 3
[Ki] = [ BA(t3/12)EABA + BB(t)EBBB] dA


I -T- -
= [ BAEABA + BBEBBB] dA


where,
EA = (t3/12)EA and EB = tEB

Then [Ki] can be rewritten as matrix equation as

follows:



-T T EA 0 BiA
[Ki] = [ BiA BiB A dA
0 E B B iB d


-T- -
= BiE Bi dA









The size of plate element stiffness will be 27 by 27.


[K] =


[ E ]
(5,5)


[ B1 B2 B3 ... Bg Bg ] dV

(5,27)


B8
B9J

(27,5)


The strain-nodal displacement matrix from which

the constant thickness is taken out is defined as [Bi].


[Bi] =


0
0
0

fi,x
fi,Y


0
- fi,y
- fi,x

0
fi


fi,x
0
fiy

fi
0


BiA]
BiB


Equivalent nodal loads due to body forces on the plate

element are calculated as


Pb = fTb dV = 1
V -1 -i


fTblJI dr ds


in which {b} = { 0 0 bz )T or ( 0 by 0 }T or { bx 0 0
T in accordance with the direction of the gravity in the

coordinate system used. The nonzero quantities bx, by, or bz

represent the body force per unit area in the direction of

application.









Equivalent loads caused by initial strains are



PO = BTEeo dV
JV
1 1
= BTEO#IJI dr ds


where,

{40)= ( xx0 Oyy0 xy 0 0 )T
= { aAT/2 aAT/2 0 0 0 T


The stresses can be calculated from the equation


[a] = [E][E]


The corresponding generalized stresses, if desired, may

be computed from



M = ( Mxy My y QM Q y T = E ( B q 40 )

It is noted that the generalized stresses are actually

moment and shear forces applied per unit length of the edge

of the plate element. Therefore these can also be turned

into common stresses using the formulation for the bending

stress calculation. The moment of inertia for the unit

length of the plate is t3 / 12. Then the in-plane stress at

a point along the thickness can be calculated as


a = Mz / I = M(t/2) / (t3/12) = 6M / t2







91


The transverse shear stresses can be found as


r =Q/ t

But this may be multiplied by a factor of 1.5 to get the

maximum shear stress at a point on a neutral surface because

the transverse shear stresses show parabolic distribution

while the calculated stresses are average stresses coming

from the assumption of a constant transverse strain along

the element z axis.













CHAPTER 5
NONLINEAR SHELL ELEMENT


5.1 Introduction



The nonlinearities included in the formulation of the

Mindlin flat shell element are those due to large

displacements and those due to initial stress

effects(geometric nonlinearity). The large displacement

effects are caused by finite transverse displacements. These

effects are taken into account by coupling transverse

displacement and membrane displacements. The initial stress

effects are caused by the stresses at the start of each

iteration. These stresses change the element stiffness for

the current iteration. These effects are evaluated directly

from the stresses at the start of each iteration and are

included in the element stiffness formation.

The total Lagrangian formulation is used. If the

updated Lagrangian formulation is used, the element

coordinate system cannot be easily formed for the next

iteration because the deformed shell is not usually planar

[26]. The symbol {} is used for a column matrix (a vector)

and the symbol [] is used for a matrix of multiple columns

and rows throughout the chapter.









5.2 Element Formulation


The generic displacements of Mindlin type shell element

are translational displacements {u v w)T and denoted as {U}.

The displacements and rotations at a point on the midplane

are (uo vo wo 9x 0y)T and denoted as (Uo}. The generic

displacements can be expressed in terms of the midplane

displacements and z as

u = Uo(x,y) + zOy(x,y)

v = Vo(x,y) zOx(x,y) (5.1)

w = Wo(x,y)

The linearized incremental strain from Eq. (2.17) is

e = ( ui,j + uj,i + uk,i uk,j + k,j k,i)
(5.2)

This equation can be written out for the strain terms to be

used for shell element using the generic displacements {u,

v, w)T


exx =
eyy=

exy =


u,

U,y
i (
2'


ez = i (



eyz = (


+ tu u, + tv, v, + tw,x ,

+ t,y U,y + t,y V,y + tw,y Wy

U,y + V,x + tU, U,y + t,x V,y + tx w,y

+ u, tuy + V,x tvy + W,x ty )

,z + w,x + tU u,z + tVx ,z + tW,x w,Z

+ u,x tu,z + V,x tv, + ,x tw,z )

V,z + w,y + tu,y u,z + t,y v,z + tWy w,z

+ u,y tU, + V,y tV, + w,y tW, )

(5.3)










The derivatives of inplane displacements u and v with

respect to x, y, and z are assumed to be small and thus the

second order terms of these quantities can be ignored

through von Karman assumption from Eqs. (2.44) [20, 21].

Furthermore the transverse displacement w is independent of

z for the shell element which means that w,z is zero. Then

Eqs. (5.3) can be reduced to

xx = u, + tw,x w,x

eyy = Uy + ty Wy

exy = ( Uy + Vx + tw,x Wy + ,x wy )

exz = ( u,z + wx )

eyz = 1 ( v,z + W,y )

(5.4)

The incremental Green's strains, sometimes called

engineering strains, can then be shown as

x = exx = Ux + tw,x w,x

4e = eyy = Uy + tw,y Wy

Ixy = 2exy = Uy + v,x + tw,x Wy + Wx ty

7xz = 2exz = u,z + wx

yz e = e = + Wy (5.5)
It is noted that the linearized nonlinear strains are left

only for inplane strain terms.

By substituting Eqs. (5.1) into Eqs. (5.5), the Green's

strain can be expressed in terms of midplane displacements.




Full Text
NONLINEAR GAP AND MINDLIN SHELL ELEMENTS
FOR THE ANALYSIS OF CONCRETE STRUCTURES
By
KOOKJOON AHN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
UNIVERSITY OF FLORIDA LIBRARIES
1990

ACKNOWLEDGEMENTS
I would like to express my deep gratitude to professor
Marc I. Hoit for his invaluable guidance and support. I also
thank professors Clifford 0. Hays, Jr., Fernando E. Fagundo,
John M. Lybas, and Paul W. Chun for being on my committee. I
also express my gratitude to professor Duane S. Ellifritt
for his help as my academic advisor at the start of my Ph.D.
program.
Thanks are also due to all the other professors not
mentioned above and my fellow graduate students, Alfredo,
Jose, Mohan, Prasan, Prasit, Shiv, Tom, Vinax, and Yuh-yi.
Finally, I am thankful to every member of my family,
especially my wife and son, for their patience and support
in one way or another.
The work presented in this dissertation was partially
sponsored by the Florida Department of Transportation.
ii

TABLE OF CONTENTS
gage,
ACKNOWLEDGEMENTS Ü
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
1.1 General Remarks 1
1.2 Link Element 2
1.3 Shell Element 5
1.4 Literature Review 5
2 GENERAL THEORIES OF NONLINEAR ANALYSIS 13
2.1 Introduction 13
2.2 Motion of a Continuum 14
2.3 Principle of Virtual Work 16
2.4 Updated Lagrangian Formulation 18
2.5 Total Lagrangian Formulation 22
2.6 Linearization of Equilibrium Equation 26
2.7 Strain-Displacement relationship
Using von Karman Assumptions 28
3 THREE-DIMENSIONAL LINK ELEMENT 34
3.1 Element Description 34
3.2 Formation of Element Stiffness 43
3.3 Solution Strategy 51
3.4 Element Verification 52
4 LINEAR SHELL ELEMENT 59
4.1 Introduction 59
4.2 Formulation of Shape Functions 59
4.3 The Inverse of Jacobian Matrix 64
4.4 Membrane Element 66
4.5 Plate Bending Element 73
5 NONLINEAR SHELL ELEMENT 92
5.1 Introduction 92
5.2 Element Formulation 93
iii

5.3 Finite Element Discretization 100
5.4 Derivation of Element Stiffness 113
5.5 Calculation of Element Stiffness Matrix .... 115
5.6 Element Stress Recovery 119
5.7 Internal Resisting Force Recovery 122
6 NONLINEAR SHELL ELEMENT PERFORMANCE 126
6.1 Introduction 126
6.2 Large Rotation of a Cantilever 126
6.3 Square Plates 133
7 CONCLUSIONS AND RECOMMENDATIONS 143
APPENDICES
A IMPLEMENTATION OF LINK ELEMENT 146
B IMPLEMENTATION OF LINEAR SHELL ELEMENT 170
C IMPLEMENTATION OF NONLINEAR SHELL ELEMENT 219
REFERENCES 230
SUPPLEMENTAL BIBLIOGRAPHY 238
BIOGRAPHICAL SKETCH 240
ÍV

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NONLINEAR GAP AND MINDLIN SHELL ELEMENTS
FOR THE ANALYSIS OF CONCRETE STRUCTURES
By
KOOKJOON AHN
August, 1990
Chairman: Marc I. Hoit
Major Department: Civil Engineering
Segmental post-tensioned concrete box girders with
shear keys have been used for medium to long span bridge
structures due to ease of fabrication and shorter duration
construction.
Current design methods are predominantly based on
linear elastic analysis with empirical constitutive laws
which do not properly quantify the nonlinear effects, and
are likely to provide a distorted view of the factor of
safety.
Two finite elements have been developed that render a
rational analysis of a structural system. The link element
is a two-dimensional friction gap element. It allows opening
and closing between the faces of the element, controlled by
the normal forces. The Mindlin flat shell element is a
combination of membrane element and Mindlin plate element.
This element considers the shear responses along the element
v

thickness direction. The shell element is used to model the
segment itself.
The link element is used to model dry joints and has
shown realistic element behavior. It opens under tension and
closes under compression. The link element has shown some
convergence problems and exhibited a cyclic behavior.
The linear Mindlin shell element to model the concrete
section of the hollow girder showed an excellent response
within its small displacement assumption.
The nonlinear Mindlin flat shell element has been
developed from the linear element to predict large
displacement and initial stress (geometric) nonlinearities.
The total Lagrangian formulation was used for the
description of motion. The incremental-iterative solution
strategy was used. It showed satisfactory results within the
limitation of moderate rotation.
Three areas of further studies are recommended. The
first is the special treatment of finite rotation which is
not a tensorial quantity. The second is the displacement
dependent loadings commonly used for shell elements. The
third is the material nonlinearity of concrete which is
essential to provide realistic structural response for safe
and cost effective designs.
vi

CHAPTER 1
INTRODUCTION
1.1 General Remarks
In the past few decades segmental post-tensioned
concrete box girders have been used for medium to long span
bridge structures. Highway aesthetics through long spans,
economy due to ease of fabrication, shorter construction
duration are some of the many advantages of precast segment
bridge construction.
The segments are hollow box sections, match cast with
shear keys in a casting yard, then assembled in place,
leaving the joints entirely dry. The shear keys are meant to
transfer service level shears and to help in alignment
during erection.
Current design methods are heavily based on linear
elastic analysis with empirically derived constitutive laws
assuming homogeneous, isotropic materials. The behavior
under load of the bridge system is very complex. Analyses
which do not properly quantify the nonlinear effects
including the opening of joints in flexure, are likely to
provide a distorted view of the factor of safety existing in
a structural system between service loads and failure. The
1

potential sliding and separation at the joints due to shear,
and by deformations generated by temperature gradients over
the depth and width of the box further complicate the
problem [1].
Two finite elements have been developed that render a
rational analysis of the system. The link element is a two-
dimensional friction gap element. It allows sliding between
the faces of the element, controlled by a friction
coefficient and the normal forces. It also accounts for zero
stiffness in tension and a very high stiffness under
compression. This link element was borrowed from rock
mechanics and newly applied to this problem to model the dry
joint between the segments. The Mindlin flat shell element
is a combination of membrane element and Mindlin plate
element. This element considers the shear responses along
the element thickness direction. The shell element was used
to model the segment itself. This element can handle large
displacement and geometric nonlinearities.
1.2 Link Element
A link element is a nonlinear friction gap element used
to model discontinuous behavior in solid mechanics. Some
examples are interfaces between dissimilar materials and
joints, fractures in the material, and planes of weakness.
These have been modeled using constraint equations, discrete

3
springs and a quasi-continuum of small thickness [2]. The
following characteristics of prototype joints were
considered.
1. Joints can be represented as flat planes.
2. They offer high resistance to compression in the
normal direction but may deform somewhat modeling
compressible filling material or crushable
irregularities.
3. They have essentially no resistance to a net tension
force in the normal direction.
4. The shear strength of joints is frictional. Small
shear displacements probably occur as shear stress
builds up below the yield shear stress.
A model for the mechanics of jointed rocks was
developed by Goodman [3], The finite element approximation
was done as a decomposition of the total potential energy of
a body into the sum of potential energies of all component
bodies. Therefore, element stiffness is derived in terms of
energy.
The Goodman element was tested for several modeled
cases.
1. Sliding of a joint with a tooth.
2. Intersection of joints.
3. Tunnel in a system of staggered blocks.
A problem with the Goodman's two dimensional model is
that adjacent elements can penetrate into each other.
Zienkiewicz et al. [4] advocate the use of continuous

4
isoparametric elements with a simple nonlinear material
property for shear and normal stresses, assuming uniform
strain in the thickness direction. Numerical difficulties
may arise from ill conditioning of the stiffness matrix due
to very large off-diagonal terms or very small diagonal
terms which are generated by these elements in certain
cases. A discrete finite element for joints was introduced
which avoids such theoretical difficulties and yet is able
to represent a wide range of joint properties, including
positive and negative dilatency (expansion and compaction
accompanying shear) [3].
The element uses relative displacements as the
independent degrees of freedom. The displacement degrees of
freedom of one side of the slip surface are transformed into
the relative displacements between the two sides of the slip
surface. This element has been incorporated into a general
finite element computer program [5]. The use of relative
displacement as an independent degree of freedom to avoid
numerical sensitivity is discussed in detail [6]. An
isoparametric formulation is given by Beer [2]. A four-node,
two-dimensional link element and a eight-node plate bending
element were used to model the dry jointed concrete box
girder bridge with shear keys [7].

5
1.3 Shell Element
The shell element is formulated through the combination
of two different elements, the membrane element and the
Mindlin plate bending element.
The Mindlin plate element is different from the
Kirchhoff plate element in that the former allows transverse
shear deformation while the latter does not.
The nonlinearities included in the formulation of the
flat shell element is for large displacement and geometric
nonlinearity due to initial stress effects. The large
displacement effects are caused by finite transverse
displacements. These effects are taken into account by
coupling transverse displacement and membrane displacements.
The initial stress effects are caused by the actual stresses
at the start of each iteration. These stresses change the
element stiffness for the subsequent iteration. These
effects are evaluated directly from the stresses at the
start of each iteration and included in the element
stiffness.
1♦4 Literature Review
The purpose of nonlinear analysis is to develop the
capability for determining the nonlinear load-deflection
behavior of the structures up to failure so that a proper

6
evaluation of structural safety can be assured. There are
two general approaches for nonlinear analysis. The first
approach is a linearized incremental formulation by reducing
the analysis to a sequence of linear solutions. The second
approach is mathematical iterative techniques applied to the
governing nonlinear equations [8].
The advantage of the incremental approach results from
the simplicity and generality of the incremental equations
written in matrix form. Such equations are readily
programmed in general form for computer solutions [9].
A generalized incremental equilibrium equation for
nonlinear analysis can be found in [10, 11, 12]. The
formulation is valid for both geometrical and material
nonlinearities, large displacements and rotations,
conservative and displacement dependent (nonconservative)
loads.
There are two frames for the description of motion. The
difference lies in the coordinate systems in which the
motion is described. These are the total Lagrangian
formulation which refers to the initial configuration [10,
11] and the updated Lagrangian formulation which refers to
the deformed configuration [12]. There have evolved two
types of notations in the description of motion. A
correlation is given these two notations, the B-notations
and the N-notations, currently used in the Lagrangian
formulation of geometrically nonlinear analysis [13]. A

short history of early theoretical development of nonlinear
analysis can be found in [9, 14].
One form of updated Lagrangian formulation is the
corotational stretch theory [15].
Shell elements are often derived from governing
equations based on a classical shell theory. Starting from
the field equations of the three-dimensional theory, various
assumptions lead to a shell theory. This reduction from
three to two dimensions is combined with an analytical
integration over the thickness and is in many cases
performed on arbitrary geometry. Static and kinematic
resultants are used. These are referred to as classical
shell elements. Alternatively, one can obtain shell elements
by modifying a continuum element to comply with shell
assumptions without resorting to a shell theory. These are
known as degenerated shell elements. This approach was
originally introduced by Ahmad, Irons, and Zienciewicz [16,
17]. Other applications can be found in [8, 18-25].
In large rotation analysis, the major problems arise
from the verification of the kinematic assumptions. The
displacement representation contains the unknown rotations
of the normal in the arguments of trigonometric functions.
Thus additional nonlinearity occurs. Further difficulties
enter through the incremental procedure. Rotations are not
tensorial variables, therefore, they cannot be summed up in
an arbitrary manner [17]. One of the special treatment of

finite rotation is that the rotation of the coordinate
system is assumed to be accomplished by two successive
rotations, an out-of-plane rotation followed by an in-plane
rotation using updated Lagrangian formulation [26, 27].
Usually the loadings are assumed to be conservative,
i.e., they are assumed not to change as the structure
deforms. One of the well known exceptions is pressure
loading which can be classified as conservative loading or a
nonconservative loading [28]. Another is the concentrated
loading that follows the deformed structure. For example, a
tip loading on a cantilever beam will change its direction
as the deformation gets larger. As loading is a vector
quantity, the change in direction means that the loading is
not conservative. Sometimes this is called a follower
loading.
The governing equation for large strain analysis can be
used for small increments of strain and large increments of
rotations [29]. This can be regarded as a generalization of
nonlinearity of small strain with large displacement. If
large strain nonlinearity is employed, an important question
is which constitutive equation should be used [9].
The degree of continuity of finite element refers to
the order of partial differential of displacements with
respect to its coordinate system. Order zero means
displacement itself must be continuous over the connected
elements. Order one means that the first order differential

of displacement must be continuous. Thus the higher order
the continuity requirement, the higher the order of assumed
displacement (shape, interpolation) function.
Mindlin-Reissner elements require only C° continuity,
so that much lower order shape functions can be used,
whereas in Kirchhoff-Love type elements, high order shape
functions must be used to satisfy the C1 continuity.
Furthermore, since Mindlin-Reissner elements account for
transverse shear, these elements can be used for a much
larger range of shell thickness. The relaxed continuity
requirements which permit the use of isoparametric mapping
techniques gives good computational efficiency if formulated
in the form of resultant stresses [30]. Unlike compressible
continuum elements, which are quite insensitive to the order
of the quadrature rule, curved C° shell elements require
very precisely designed integration scheme. Too many
integration points result in locking phenomena, while using
an insufficient number of quadrature points results in rank
deficiency or spurious modes [30]. While Gauss point stress
results are very accurate for shallow and deep, regular and
distorted meshes, the nodal stresses of the quadratic
isoparametric Mindlin shell element are in great error
because of the reduced integration scheme which is necessary
to avoid locking [31].
The degenerate solid shell element based on the
conventional assumed displacement method suffers from the

locking effect as shell thickness becomes small due to the
condition of zero inplane strain and zero transverse shear
strain. Element free of locking for linear shell analysis
using the formulation based on the Hellinger-Reissner
principle with independent strain as variables in addition
to displacement is presented in [32].
Shear locking is the locking phenomenon associated with
the development of spurious transverse shear strain.
Membrane locking is the locking phenomenon associated with
the development of non-zero membrane strain under a state of
constant curvature. Machine locking is the locking
phenomenon associated with the different order of dependence
of the flexural and real transverse shear strain energies on
the element thickness ratio, and it is therefore strictly
related to the machine finite word length [33]. Some of the
solutions are as follows:
1. Assumed strain stabilization procedure using the Hu-
Washizu or Hellinger-Reissner variational principles
[33].
2. The assumed strain or mixed interpolation approach [34,
35] .
3. Suppressing shear with assumed stress/strain field in a
hybrid/mixed formulation [30]. Suppression of zero
energy deformation mode using assumed stress finite
element [36].
4. Coupled use of reduced integration and nonconforming
modes in quadratic Mindlin plate element [37].
5. Higher order shallow shell element, with 17 to 25 nodes
[38, 39].

6. Global spurious mode filtering [40].
7. Artificial stiffening of element to eliminating zero
energy mode, special stabilizing element [41].
In the faceted elements, due to the faceted
approximation of the shell surface, coupling between the
membrane and the flexural actions is excluded within each
individual element, the coupling is, however, achieved in
the global model through the local to global coordinate
transformation for the elements [39].
In geometrically nonlinear analysis with flat plate
elements, it is common to use the von Karman assumptions
when evaluating the strain-displacement relations. The
assumption invoked is that the derivatives of the inplane
displacements can be considered to be small and hence their
quadratic variations neglected. However, this simplification
of the nonlinear strain-displacement relationship of the
plate, when used in conjunction with the total Lagrangian
approach, implies that the resulting formulation is valid
only when the rotation of the element from its initial
configuration is moderate. Thus for the total Lagrangian
approach to handle large rotations, simplifications of the
kinematic relationship using the von Karman assumptions is
not permitted [39].
Some of the special solution strategies to pass the
limit point are given in references [25, 42-48]. A limit
point is characterized by the magnitude of tangential

stiffness. It is zero or infinite at a limit point. Thus
conventional solution strategies fail at the limit point.
Arc length method was introduced in reference [42], and
applied in the case of cracking of concrete [43], This was
improved with line search and accelerations in references
[44, 45]. Line search means the calculation of an optimum
scalar step length parameter which scales the standard
iterative vector. This can be applied to load and
displacement control and arc length methods [44].
The traditional solution strategies are iterative
solutions, for example, Newton-Raphson, constant stiffness,
initial stiffness, constant displacement iteration, load
increment [46] along with Cholesky algorithm with shifts for
the eigensolution of symmetric matrices [47] for element
testing for spurious displacement mode.
The vector iteration method without forming tangent
stiffness for the postbuckling analysis of spatial
structures is also noted [48].
The linearized incremental formulation in total
Lagrangian description has been used for this study of large
displacement nonlinearity including initial stress effects.
The special treatment of finite rotation is not included in
the current study. Material nonlinearity is also excluded.

CHAPTER 2
GENERAL THEORIES OF NONLINEAR ANALYSIS
2.1 Introduction
The incremental formulations of motion in this chapter
closely follow the paper by Bathe, Ramm, and Wilson [11].
Other references are also available [9, 10, 12, 14, 15, 49,
50, 51].
Using the principle of virtual work, the incremental
finite element formulations for nonlinear analysis can be
derived. Time steps are used as load steps for static
nonlinear analysis. The general formulations include large
displacements, large strains and material nonlinearities.
Basically, two different approaches have been pursued
in incremental nonlinear finite element analysis. In the
first, Updated Lagrangian Formulation, static and kinematic
variables, i.e., forces, stresses, displacements, and
strains, are referred to an updated deformed configuration
in each load step. In the second, Total Lagrangian
Formulation, static and kinematic variables are referred to
the initial undeformed configuration.
It is noted that using either of two formulations
should give the same results because they are based on the
13

14
same continuum mechanics principles including all nonlinear
effects. Therefore, the question of which formulation should
be used merely depends on the relative numerical
effectiveness of the methods.
2.2 Motion of a Continuum
Consider the motion of a body in a Cartesian coordinate
system as shown in Fig. 2-1. The body assumes the
equilibrium positions at the discrete time points 0, dt,
2dt, ..., where dt is an increment in time. Assume that the
solution for the static and kinematic variables for all time
steps from time 0 to time t, inclusive, have been solved,
and that the solution for time t+dt is required next.
The superscript on left hand side of a variable shows
the time at which the variable is measured, while the
subscript on left hand side of a variable indicates the
reference configuration to which the variable is measured.
Thus the coordinates describing the configuration of the
body using index notation are
At time 0 = °x¿
At time t = ^x¿
At time t+dt = t+dtxi

15
Fig. 2-1 Motion of a Body

The total displacements of the body are
At time 0
At time t
u j
'Uj
At time t+dt =
_ t+dt
Uj
The configurations are denoted as
At time 0 = °C
At time t = UC
=
At time t+dt = t+dt(
Thus, the updated coordinates at time t and time t+dt are
^ = °Xi + ^
t+dtx. = Ox. + t+dtu.
The unknown incremental displacements from time t to
time t+dt are denoted as (Note that there is no superscript
at left hand side.)
u. = t+dtu.
'U:
(2.1)
2.3 Principle of Virtual Work
Since the solution for the configuration at time t+dt
is required, the principle of virtual work is applied to the
equilibrium configuration at time t+dt. This means all the
variables are those at time t+dt and are measured in the
configuration at time t+dt and all the integrations are
performed over the area or volume in the configuration at
time t+dt. Then the internal virtual work (IVW) by the

corresponding virtual strain due to virtual displacement in
t+dtC is
IVW =
t+dt ..
t+dt
s
' t+dt -..
t+dt 13
•
-
-
(t+dt dv)
(2.2)
where,
t+dt . ,
t+dt x3
Stresses at time t+dt measured in the
configuration at time t+dt.
= Cauchy stresses.
= True stresses.
*;+d^ eji = Cauchy's infinitesimal(linear) strain tensor
t+dt referred to the configuration at time t+dt.
= Virtual strain tensor.
S
Delta operator for variation.
and the external virtual work (EVW) by surface tractions and
body forces is
EVW =
t+dt x.
£
t+dt
t+dt k
t+dt k
0
_
(t+dt dAj
where,
t+dt a.
t+dt k
t+dt
t+dt
uk
t+dt
t+dt b
X
t+dt u
t+dt k
t+dt p
t+dt k
0
_
(t+dt dV)
(2.3)
Surface traction at time t+dt measured in
the configuration at time t+dt.
Total displacement at time t+dt measured in
the configuration at time t+dt.

6 *"+u^ = Variation in total displacement at time
+ t+dt measured in configuration at time t+dt
= Virtual displacement.
= Mass density per unit volume.
= Body force per unit mass.
and all the integration is performed over the area and the
volume at time t+dt.
2.4 Updated Laqranqian Formulation
t+dt
t+dt 9
t+dt H
t+dt bk
In this formulation all the variables in Eqs. (2.2) and
(2.3) are referred to the updated configuration of the body,
i.e, the configuration at time t. The equilibrium position
at time t+dt is sought for the unknown incremental
displacements from time t to t+dt.
The internal virtual work, the volume integral in Eq.
(2.2) measured in the configuration at time t+dt can be
transformed to the volume integral measured in the
configuration at time t in a similar manner that is given in
reference [52]
IVW
t+dt
t+dt ij
t+dt
t+dt
(t+dt dv)
t+dt c
t Sij
t+dt
t
e
ij
(fc dV) = EVW (2.4)

where,
t+dt s _ second Piola-Kirchhoff (PK-II) stresses
t J measured in the configuration at time t.
S eji = Variations in Green-Lagrange (GL) strain
tensor measured in the configuration at
time t.
The PK-II stress tensor at time t+dt, measured in the
configuration at time t can be decomposed as
t+dt e
t Sij
ID
tsij
(2.5)
because the second PK-II stress at time t measured in the
configuration at time t is the Cauchy stress.
From Eq. (2.1), the total displacements at time t+dt
measured in the configuration at time t is
t+dt Ui = t u. + ^ u. „ t u. (2.6)
This is true because the displacement at time t measured in
the configuration at time t is zero. In other words, the
displacement at time t+dt with respect to the configuration
at time t is the incremental displacement itself.
And the GL strain is defined in terms of displacement as
E
ij = *
(2.7)
E and U are used in the places of e and u to avoid confusion
between general strain and incremental strain, and between

20
general displacement and incremental displacement used in
this formulation. It is noted that these finite strain
components involve only linear and quadratic terms in the
components of the displacement gradient. This is the
complete finite strain tensor and not a second order
approximation to it. Thus this is completely general for any
three-dimensional continuum [52].
Then the GL strain tensor at time t+dt measured at time
t can be calculated as
t+dt
t
+ {(-tuk + ^tUk +
(2.8)
where
teij teij + t^ij
Incremental GL strain in **C
Linear portion of incremental GL strain in **C
This is linear in terms of unknown incremental
displacement.
Linearized incremental GL strain in ^C.

21
t»?ij i tUjcOJ
= Nonlinear portion of incremental GL strain.
The variations in Green-Lagrange strain tensor at time
t+dt measured in the configuration at time t can be shown as
using Eg. (2.8) .
8 t+dt £ij 8 ( t^ij +tei3 J 8 teij
(2.9)
4» 4“ t t
S^elj = 0 because is known. There is no variation m
known quantity.
Then using the Eqs. (2.5), (2.8) and (2.9), the
integrand of Eq. (2.4) becomes
t+dt s. s t+dt e. . _ ^ t
'i: t
ID
+ Si j ) 6 . e.
t 'ID t ID ' " t ID
" (trij + tSiD)(5 teiD + 5t'?ij)
=tSiD(5teiD + 5t'?iD) + trij 5 teiD + triD 6 t^D
=tSiD 5 teiD + tfij 8 teiD + t rij 8 t^ij (2.10)
The constitutive relation between incremental PK-II
stresses and GL strains are
tSij tCiDkj tCkl
(2.11)

22
Finally the equilibrium Eq. (2.4) from the principle of
virtual work using Eqs. (2.10) and (2.11) is
tcijkl tekl 8 t€ij
fcdV
+
trij
'dV
= EVW
(2.12)
where, the external virtual work must be transformed from
t+dtc to tC. This is not applicable to conservative loading,
i.e., loading that is not changed during deformation.
EVW
t+dt
t
t+dt
t
uk
(fcdA)
t+dt „
t p
t+dt y.
t bk
s
t+dt
t Uk
-
(fcdV)
(2.13)
and this is the general nonlinear incremental equilibrium
equation of updated Lagrangian formulation.
2.5 Total Lagrangian Formulation
Total Lagrangian formulation is almost identical with
the updated Lagrangian formulation. All the static and
kinematic variables in Eqs. (2.2) and (2.3) are referred to

the initial undeforxned configuration of the body, i.e, the
configuration at time 0. The terms in the linearized strain
are also slightly different from those of updated Lagrangian
formulation.
The volume integral in Eq. (2.2) measured in the
configuration at time t+dt can be transformed to the volume
integral measured in the configuration at time 0 as [52]
t+dt , ,
t+dt XD
t+dt e..
t+dt iD
(t+dtdv)
t+dt
t+dt
ID
(°dV)
(2.14)
where,
t+dtg _ second Piola-Kirchhoff stress tensor
° measured in the configuration at time 0.
5 t+dt£ _ variations in Green-Lagrange (GL) strain
° tensor measured in the configuration at
time 0.
The PK-II stress tensor at time t+dt, measured in the
configuration at time 0 can be decomposed as
t+dt
o
(2.15)
From Eq. (2.1), the total displacements at time t+dt
measured in the configuration at time 0 is

24
t+dtu, = tu,
u.
(2.16)
Then the GL strain tensor at time t+dt measured at time
0 can be calculated as
t+dt.
3 ‘ v q“3 ' 0U j ^ ' i
O 'ij - i [(^ + 0Ui)(j + (V
+< ^1*1 "1 t Í/Í *1
o XJ o XJ o +3
Oeij + oeij
where,
t
o€ij
oeij
eii
o XJ
o’ij
= 1
(tui -A + tu 4 + tuk 1 tuk -¡)
= GL strain at time t in °C.
ei-i + 'ii-i
o +J o XJ
(2.17)
= Incremental GL strain in °C.
= | ( \li -s + Uj i + tUl_ • u> + tu . u> i )
= Linear portion of incremental GL strain in °C.
= This is linear in terms of unknown incremental
displacement.
= Linearized incremental GL strain in °C.
" i
= Nonlinear portion of incremental GL strain.
The variations in Green-Lagrange strain tensor at time
t+dt measured in the configuration at time t can be shown as

using Eq. (2.17).
c t+dt
0 €
ID
5 < o‘ij + oij > = 5 Sij
(2.18)
S = 0 because te-!-i is known. There is no variation in
o 1D o
known quantity.
Then using the Eqs. (2.15), (2.17) and (2.18), the
integrand of Eq. (2.14) becomes
t+dt
o
sii 6 t+dtfii = ( tsii + sii ) «
J-D o 1D o 1D o 1D o 1D
■ (oSij + (5 oeij + 5 o’ii1
=oSij(i oeiJ + * O "ij ’ +otsij Soeij +oSij S
= S e-s-t + 5 S rii-: (2.19)
o XD o 1D o -LD o 1D o ^-J o 1D
The constitutive relation between incremental PK-II
stresses and GL strains are
*ij oCijkj 0ekl
(2.20)
Finally the equilibrium Eq. (2.14) from the principle
of virtual work using Eqs. (2.19) and (2.20) is

26
en 5 e-s-í °dV
'ijkl 0ekl 0 0eij
tsi-i 5 «íi-i °dv
o -LJ O XJ
= EVW -
tsii S e^ °dV
o 1J o +J
(2.21)
where, the external virtual work must be transformed from
t+dtC to °C. This is not applicable to conservative loading,
that is, loading that is not changed during deformation.
EVW =
1
X
-p
-p
+
•P 0
1
6
' t+dt u 1
o K
•
L J
-
(°dA)
t+dt
o p
t+dt bk 1
o K
6
t+dt ..
uk
O K
•
-
(°dV)
(2.22)
and this is the general nonlinear incremental equilibrium
equation of total Lagrangian formulation.
2.6 Linearization of Equilibrium Equation
The incremental strain from time t to t+dt is assumed
to be linear, i.e., ekl = ekl in Eqs. (2.11), (2.12),
(2.20), and (2,21).

27
For the updated Lagrangian formulation,
tSij ' tCijkj tekl
(2.23)
and,
tcijkl tekl 5 teij dv
t'U s t’ij tdV
= EVW -
trij 5 teij tdV
(2.24)
For the total Lagrangian formulation,
oSij 0Cijkj o6kl
(2.25)
and,
Cijkl
ekl ^ ein
O Jvx o J-J
3dV
+
S
o^ij
°dV
= EVW
(2.26)
It should be noted that the surface tractions and the
body forces in the calculation of external virtual work may
be treated configuration dependent when the structure
undergoes large displacements or large strains. If this is

the case, the external forces must be transformed to the
current configuration at each iteration [10, 11, 12].
2.7 Strain-Displacement Relationship
Using the von Karman Assumptions
The nonlinear strain terms can be simplified for the
plate or shell type structures using von Karman assumption
of large rotation.
In the mechanics of continuum the measure of
deformation is represented by the strain tensor E^j [52] and
is given by using index notation.
2Eij = ( ui,j + uj,i + uk,iuk,j ) (2.2*7)
where,
u¿ = Displacement in i-direction.
ui,j = aui / axj
x¿ = Rectangular Cartesian coordinate axes, i=l,2,3.
uk,iuk,j = ul,iul,j + u2,iu2,j + u3,iu3,j
The von Karman theory of plate is a nonlinear theory
that allows for comparatively large rotations of line
elements originally normal to the middle surface of plate.
This plate theory assumes that the strains and rotations are
both small compared to unity, so that we can ignore the
changes in geometry in the definition of stress components
and in the limits of integration needed for work and energy

considerations [53]. It is also assumed that the order of
the strains is much less than the order of rotations.
29
If the linear strain e^j and the linear rotation r^j
are defined as
(2.28)
2eij = ui,j + uj,i
(2.29)
Then the sum of Eqs. (2.28) and (2.29) gives
(2.30)
and the subtraction of Eq. (2.29) from Eq. (2.28) gives
(2.31)
From Eqs. (2.30) and (2.31), it is concluded that
(2.32)
uk,j “ ekj + rkj
uk,i = eik " rik
(2.33)
Eq. (2.33) can be rewritten as
(2.34)
uk,i “ eki + rki
since e^^ = e^ from the symmetry of linear strain terms and
r^ = -r^ from the skew symmetry of the linear rotation
terms.
The strain-displacement Eq. (2.27) now becomes
2Eij = 2eij + (eki + rki)(ekj + rkj)
(2.35)

by substituting Eqs. (2.30) through (2.34) into Eq. (2.27).
Thus the nonlinear strain terms have been decomposed into
linear strain terms and linear rotation terms.
From the assumption on the order of strains and
rotations
eki << rki and ekj « rkj (2.36)
Thus Eq. (2.35) can be simplified as by ignoring ek¿ and
ekj'
2Eij = 2e¿j + rkirkj (2.37)
The straight line remains normal to the middle surface
and unextended in the Kirchhoff assumption, but it is not
necessarily normal to the middle surface for the Mindlin
assumption. For both assumptions the generic displacements
u,v,w can be expressed by the displacements at middle
surface.
For the Kirchhoff plate [20],
u(x,y,z) = uQ(x,y) - z[wQ(x,y),x]
v(x,y,z) = vQ(x,y) - z[wQ(x,y),y] (2.38)
w(x,y,z) = wQ(x,y)
where,
uQ, Vq, W0 = Displacements of the middle surface
in the direction of x, y, z.
u, v, w = Displacements of an arbitrary point
in the direction of x, y, z.

31
Now the linear strain components e^j and the linear rotation
components r^j can be calculated using Eqs. (2.28) and
(2.29).
ell
=
1
2
(ul,l
+
ul,l)
= ul,l = U'X
e12
=
1
2
(ul,2
+
u2,l)
= I(u,y + v,x
e13
—
1
2
(ul,3
+
u3,l)
= *(~wo'x + w
e22
=
1
2
(u2,2
+
u2,2>
= u2,2 = v'y
0)
to
to
=
1
2
(u2,3
+
u3,2)
= H-w0,y + w
e33
=
1
2
(u3,3
+
u3,3)
= u3,3 = 0
(2.39)
The rotation terms r12, ri3' r23 are the rotation quantities
about the axes 3(z), 2(y) and l(x), respectively. For the
plate located in the xy plane, the rotation about z axis r12
is much smaller than rotation about x axis r23 and y axis
r13 and therefore r12 is assumed to be zero here. And it is
noted further that wQ(x,y) is the same as w(x,y) and is a
function of only x and y so that w,3 = w,z = 0.
r12I « Ir23I or Ir13I (2.40)
rll - 1(ulfl
- u3 x)
= 0
r12 = *(ul,2
“ u2,l)
= Hu,y -
V,x) = 0
r13 = Hu1/3
“ u3,l)
= H-wD,x
- w,x) = -w,x
r22 = Hu2/2
“ u2, 2)
= 0
r23 = = iu2,3
“ u3,2)
= H-w0,y
- w/y) = -w,y
r33 = *(u3,3
" u3, 3)
= 0
(2.41)

32
The linear strain component e^j is symmetric and the linear
rotation component r^j is antisymmetric.
eij = eji
rij — -rj^ (2.42)
The strain components from Eq. (2.37) can be rewritten using
Eqs. (2.39) and (2.41).
E
E
E
E
E
E
XX
=
en
+
1
2
yy
=
e22
+
1
2
zz
=
e13
+
1
2
xy
=
e12
+
1
2
(rllr12
xz
=
e13
+
1
2
(rllr13
yz
=
e23
+
A
2
(r12r13
2
21
+
r312)
= e1]L + |r31
2
22
+
r322)
= e22 + ^r32
2
23
+
r332)
= 3(r132 + r
+ r21r22 + r31r32> = e12 + "r31r32
+ r21r23 + r31r33)
+ r22r23 + r32r33^ (2.43)
Ezz term is assumed to be zero because it does not have the
linear term. Exz and EyZ terms are transverse shear terms
which can be ignored for thin plate. Then Eq. (2.43) can be
rearranged as follows using Eqs. (2.41) if all the zero
terms are removed.
Exx =
ell
+
i— 2
2 r31
= ell
+
1
2
(w,x)2
II
>i
>i
w
e22
+
2 r322
= e2 2
+
1
2
CM
'lx
Exy
e12
+
2r31r32
= e12
+
1
2
(W,x) (w/y)
(2.44)
Exz = e13
Eyz = e23

Thus the decomposition of exact strain components has been
done using the Kirchhoff plate assumptions (2.38) and the
von Karman assumption (2.40) on the magnitude of rotation.
It is noted that all the inplane displacement gradients in
nonlinear strain terms are ignored through von Karman
assumptions [20], This fact will be applied in chapter 5.

CHAPTER 3
THREE DIMENSIONAL LINK ELEMENT
3.1 Element Description
The link element used here is based on the two
dimensional element developed by Cleary [54]. The link
element is based on the following assumptions.
Any normal compressive force is transferred to the
other side of the link without any loss. To facilitate this,
a very limited amount of loss through displacement should be
allowed. Currently, this limited displacement is defaulted
to .001 units, while it is a input parameter. The link
separates in response to any net tension, losing its normal
stiffness.
To discuss the shear force transfer, some definitions
for friction are needed. The force to start one body sliding
along the other body is called the static friction force.
The force to keep it moving is the kinetic friction force.
There are two corresponding coefficients of friction, static
friction coefficient and dynamic friction coefficient, where
the static friction coefficient will generally be greater
than the dynamic friction coefficient.
34

Two laws of friction were used in the link element. The
first law is that the frictional force is proportional to
the normal force, with the constant of proportionality being
the friction coefficient. The second law is that friction
does not depend on the apparent area of the connecting
solids, i.e., it is independent of the size of the bodies.
The shear force is transferred through friction. The
uncertainty in friction is the factor which limits the
overall accuracy of the calculation. Therefore, it is
assumed that the static friction coefficient is proportional
to the dynamic friction coefficient. For nonmetallic
materials, the ratio of dynamic coefficient to static
coefficient is about 0.75.
The link element is composed of two surfaces. If the
shear force is less than or equal to the static friction
force, i.e., coefficient of friction times the normal force,
the shear force is balanced by the friction force and the
total force is transferred. This is shown in Fig. 3-1. But
if the shear force is greater than the static friction
force, one surface of the link element will move along the
other surface. In this case there will be a dynamic friction
force which is less than the shear force. This dynamic
friction force can only resist a portion of the shear and
the system is not in static equilibrium. Therefore, if the
shear force is greater than the static friction force, the
link element will lose its shear stiffness. This can also be

modeled with a body on roller and spring as shown in Fig. 3-
2. The spring model of the link element is shown in Fig. 3-
3.
The link element here has four nodes and each node has
three translational degrees of freedom in local u-, n-, and
w-directions. The total number of element degrees of freedom
is 12. The element degrees of freedom are shown in Fig. 3-4.
The equivalent "strain" for the link element is defined
as the average deformation at the center of the element. The
average deformation corresponding to the translational
degrees of freedom, i.e., uQ, vQ, and wQ, can be directly
calculated from the joint displacements by averaging the
difference in nodal displacements at the ends of element in
local u-, n-, and w-direction in turn. The relative rotation
at the center of the element, rQ, can be found using nodal
displacements in local n-direction and the element length as
shown in Fig. 3-5. This angle is not an "average" value but
the "relative" change in angle of the center line due to
rotation.
The two joint parameters must be introduced. These are
kn, the unit stiffness normal to the joint, and ks, the unit
stiffness along the joint.
The off-diagonal term kns to account for dilatation
during shearing is ignored because this joint element will
model the dry joint between concrete box girder bridge
segments. No significant dilation is expected in this case.

Some values of kn and ks were reported in geotechnique
area [3]. As the values are those for natural joints, they
do not directly apply to this case.
From the test results [7], it can be seen that the
shear stiffness of dry joint ranges from 70,000 to 286,000
psi per inch at ultimate. In case of frictional strength,
this can be interpreted as linear behavior between the
origin and the ultimate point.
It seems reasonable that the normal stiffness of the
element, kn, is assumed to be stiffer than the connected
material by the order of 103 to transfer the normal force
without any significant loss. The forces are either totally
transferred in compression or totally lost in tension. The
latter case has no problem related to the value of kn.
The shear stiffness parameter is more difficult to
define. The data available is so limited that even a
statistical treatment cannot be done. But in the analysis of
structural behavior up to the ultimate, these properties do
not have great influence because the forces are transferred
through friction.
The shear stiffness becomes zero upon sliding. But
there is some 'residual' shear force. This 'residual' force
is equal to friction force. Therefore, if shear displacement
is more than the displacement just before the sliding the
shear stiffness is set to be zero.

38
N
F = Friction force,
f
m = Friction coefficient
1) P < or = mN then P = Friction Force. In Equilibrium.
2) P > mN then the body moves but the frictional force
mN is acting against the other body.
Fig. 3-1 Friction Force

FRICTIONAL SPRING WITH SHEAR STIFFNESS
BEFORE SLIDING
F
FRICTIONAL SPRING WITH ZERO STIFFNESS
mN < F
AFTER SLIDING
Fig. 3-2 Spring Model of Friction Force

40
SPRING MODEL FOR SHEAR FORCES
Fig. 3-3 Spring Model of Link Element

n
A11
Ik É.
i
^8
’
u
w
Fig. 3-4 Element Degrees of Freedom of Link Element

Fig. 3-5 Element "Strain"

3.2 Formation of Element Stiffness
There are four nodes per element. Each node has three
degrees of freedom corresponding the translational
displacements in u-, n-, and w-direction resulting in 12
element degrees of freedom as shown in Fig. 3-4. The element
stiffness is derived directly from the physical behavior of
the element described in section 3.1.
The mathematical symbol {} is used for a column vector
and [] for a matrix. The nodal displacement column vector
{q}(12) is composed of 12 translational nodal displacements
corresponding to the 12 element degrees of freedom.
{q} = { u± V;L Uj Vj Wj uk vk wk Ul wx }T
The "strain” is defined as the average deformation at
the center of the element as shown in Fig. 3-5. The "strain"
column vector (e}(4) is
{e} = { uQ vQ wQ rQ }T
where,
uo = (
uk
+
U1
)
/
2 -
( Uj.
+
uj
)
/
2
vo = (
vk
+
V1
)
/
2 -
(vi
+
vj
)
/
2
wo = (
wk
+
W1
)
/
2 -
( wA
+
Wj
)
/
2
ro = (
vk
-
V1
)
/
L -
( Vj
-
vi
)
/
L
where,
L = The length of the element.

44
uQ/ v0, wq = Average nodal displacements in local
u-, n-, w-directions.
rQ = The relative angle change about local z axis.
Therefore the relationship between "strain" and nodal
displacements is
{e>(4) = [B](4,12) {q}(12)
The [B](4,12) matrix which gives strains due to unit
values of nodal displacements is
-0.5
0.0
0.0
-0.5
0.0
0.0
0.5
0.0
0.0
0.5
0.0
0.0
0.0
-0.5
0.0
0.0
-0.5
0.0
0.0
0.5
0.0
0.0
0.5
0.0
0.0
0.0
-0.5
0.0
0.0
-0.5
0.0
0.0
0.5
0.0
0.0
0.5
0.0
1/L
0.0
0.0
-1/L
0.0
0.0
1/L
0.0
0.0
-1/L
0.0
The "stress" is defined as the normal and shear stress
per unit of area. {s> is the average stress on the surface
due to the two nodal forces exerted in the plane of the
surface. This stress is in equilibrium with the stress on
the other surface of the element as shown in Fig. 3-6. m is
the moment of the nodal forces on one surface in local n-
direction about the center of the element. This moment is
also balanced by the moment of the nodal forces on the other
surface of the element. This moment is used to define the
distribution of the normal stress of the element as shown in
Fig. 3-7.

45
Local Coordinate System
Fig. 3-6 Nodal Forces and Stresses of Link Element

FORCE TRANSFER THROUGH ONE EDGE
OF THE LINK ELEMENT
Fig. 3-7 Element "Strain", m

The "stress" column vector (s}(4) is
{s} = { sx, sn, sz, m }
The "stress-strain" relationship is
{s}(4) = [E](4,4) {e}(4)
where,
0
0
0
V
where can be related to kn using the definition of
the moment m, i.e.,
m = (sn)(L)(t)(0.5)(L)
= *m*0 = ^(Vf0-5)^))
Thus, kjn = (knVD) (L) (t) (0.5) (L) / [V0/ (0.5) (L) ]
= (0.25) (t) (kn) (L3)
where, t = Element thickness.
This assumes that there is no coupling between the
shear stress and normal stress.
The element nodal force column matrix (P}(12) is
composed of the 12 nodal forces shown in Fig. 3-6.

48
{P} { Pui Pni pwi puj pnj pwj puk pnk pwk
pul pnl Pwl >T
Stress can then be related to nodal forces using the
definition of stress and force equilibrium between the two
surfaces of the element.
By the definition of stress,
sn = (1/Lt)( Pnk + Pnl ) (3.1)
sx = (1/Lt)( Puk + Pul ) (3.2)
sz = (1/Lt)( Pwk + Pwl ) (3.3)
m =Pnk(0.5)(L) -Pnl(0.5)(L) (3.4)
where, Lt = (L)(t)
By force equilibrium of the two surfaces,
P¿ = -P1 and Pj = -Pk (3.5)
To express the element nodal forces in terms of the
stress, we use Eqs. (3.1) through (3.5) to find the force
recovery matrix [FR]. [FR] gives the nodal forces in
equilibrium with the element stresses.
From (Eq. (3.1) + Eq. (3.4)),
2Pnk = (L)(t)(sn) + 2(m)/L
Pnk = 0.5(L)(t)(sn) + (1/L)(m)

49
From Eq. (3.5),
pnj = ~pnk
= -0.5(L)(t)(sn) - (1/L)(m)
From Eq. (3.1),
Pnl = (L)(t)(sn) - Pnk
= (L)(t)(sn) - ( 0.5(L)(t)(sn) + (1/L)(m))
= 0.5(L)(t)(sn) - (1/L)(m)
From Eq. (3.5),
pni = "pnl
= -0.5(L)(t)(sn) + (1/L)(m)
From the assumption that Puk = Pu^ and Eq. (3.2),
Puk = (L)(t)(sx)/2
Pul = (L)(t)(sx)/2
From Eq. (3.5),
Pui = - pul = "(L)(t)(sx)/2
Puj = - Puk = -(L)(t)(sx)/2
From the assumption that Pwk = Pw^ and Eq. (3.3),
Pwk = (L)(t)(sz)/2
PW1 = (L)(t)(sz)/2

50
From eqn 5,
pwi = - pwl = "(L> (tHSz)/2
pwj = " pwk = "(L)(t)(sz)/2
Therefore, the force-stress relationship is
{P}(12) = [FR](12,4) {s}(4)
where the force recovery matrix [FR](12,4) is
-Lt/2
0
0
0
0
-Lt/2
0
1/L
0
0
-Lt/2
0
-Lt/2
0
0
0
0
-Lt/2
0
-1/L
0
0
-Lt/2
0
Lt/2
0
0
0
0
Lt/2
0
1/L
0
0
Lt/2
0
Lt/2
0
0
0
0
Lt/2
0
-1/L
0
0
Lt/2
0
And this relationship is further expanded using the
stress-strain relationship and the strain-nodal displacement
relationship as follows.
{P >(12) = [FR](12,4) [E](4,4) (e)(4)
= [FR](12,4) [E](4,4) [B](4,12) (q}(12)
= [Bt](12,4) [E](4,4) [B](4,12) (q)(12)
Then finally this can be symbolized as equilibrium
equation.

51
{P>(12)— [Ke](12,12) {q}(12)
where [Ke] = [Bt][E][B]
Here it is noted that [FR] = [Bt] and [Ke] = [Bt][E][B] just
as in the case of common finite element method.
The final element stiffness matrix [Ke] is
kx
0
0
kx
0
0
-kx
0
0
-kx
0
0
0
2kn
0
0
0
0
0
0
0
0
-2kn
0
0
0
kz
0
0
kz
0
0
-kz
0
0
-kz
kx
0
0
kx
0
0
-kx
0
0
-kx
0
0
0
0
0
0
2kn
0
0
-2kn
0
0
0
0
0
0
kz
0
0
kz
0
0
-kz
0
0
-kz
-kx
0
0
-kx
0
0
kx
0
0
kx
0
0
0
0
0
0
-2kn
0
0
2kn
0
0
0
0
0
0
-kz
0
0
-kz
0
0
kz
0
0
kz
-kx
0
0
-kx
0
0
kx
0
0
kx
0
0
0
-2kn
0
0
0
0
0
0
0
0
2kn
0
0
0
-kz
0
0
-kz
0
0
kz
0
0
kz
This matrix can be rotated to any direction using the
standard rotation.
3.3 Solution Strategy
The structural stiffness changes because of the slip
and debonding of the link. Therefore, the process of the
resistance of the total structure physically becomes
nonlinear. Correspondingly, special solution techniques for
nonlinear behavior are needed.
This can be done using the iterative solution technique
with initial stiffness or tangent stiffness. The latter can

be formed by assembling the structural stiffness at the
beginning of each iteration and this converges faster than
the initial stiffness.
A third solution strategy for this case is event-to-
event technique which is usually employed for the linear
stiffnesses between any two "events,” which are defined as
the intersection point between two linear segments. This
also provides means of controlling the equilibrium error.
Any significant event occurring within any element
determines a substep. The tangent stiffness is modified in
each substep, and hence, the solution closely follows the
exact response.
3.4 Element Verification
3.4.1 SIMPAL
The finite element analysis program SIMPAL [55], is
used to implement and verify the element formulation. SIMPAL
was chosen for the initial implementation because that was
the original implementation done by Cleary [54]. This way,
the 3-D aspects could be implemented and verified using
Cleary's original program. A table of the element
verification is shown in Fig. 3-8 and Fig. 3-9.

LOADING
RESULTS
NODE
THEORY
SIMPAL
ERROR
DISP
2
-.1333
-.1333
.000
DISP
4
-.1333
-.1333
.000
STRESS
N/A
-80
-80
.000
NODE
THEORY
SIMPAL
ERROR
DISP
2
-.1333
-.1333
.000
DISP
4
-.1333
-.1333
.000
STRESS
N/A
-80
-80
.000
NODE
THEORY
SIMPAL
ERROR
DISP
2
-.1333
*
-.1337
.003
DISP
4
-.1333
★
-.1337
.003
STRESS
N/A
-80
-80
.000
THICKNESS = .25
Ks = 3E6
Kn = 6E6
NODE 2
NODE 4
YDISP =-0.1017-04 Z DISP =-0.8684-05
Z DISP = -0.8684-05 Y DISP = -0.1017-04
2 2
SQRT((.1017) + (.08684) ) = 0.1337
Fig. 3-8 Link Element Test Using SIMPAL

54
LOADING
RESULTS
Z
Fig. 3-9 Combined Test Model for SIMPAL

55
3.4.2 ANSR
The test examples used are the same as those used in
the initial element verification using SIMPAL. The results
from ANSR [56] are exactly the same as those from SIMPAL.
The link element was tested further using a modeled membrane
element composed of 22 truss elements as a membrane element
was not available at the time of element verification in
ANSR. The results are shown in Table 3-1 and the structures
used are shown in Fig. 3-10 and Fig. 3-11.

Table 3-1 Displacements of Truss Model
for ANSR
Node
No.
Truss
only
Truss
w/ LINK
Diff.
(%)
10-x
-.1027e-4
-.1049e-4
2.2
10-y
-.1990e-5
-.2010e-5
0.9
11-x
-.9017e-5
-.9211e-5
2.2
11-y
-,4906e-6
-.4973e-6
1.4
12-x
-.9915e-5
-.1049e-4
2.2
12-y
+.9742e-6
+.9703e-6
0.4

57
Fig. 3-10 Combined Test Model for ANSR

58
Fig. 3-11 Truss Model for ANSR

CHAPTER 4
LINEAR SHELL ELEMENT
4.1 Element Description
The shell element is formulated through a combination
of two different elements, the membrane element and the
Mindlin plate bending element [57].
The Mindlin plate element is different from the
Kirchhoff plate element in that the former allows transverse
shear deformation while the latter does not.
The common portions of the formulation of two elements
are
1. Formation of the shape functions.
2. Formation of the inverse of Jacobian matrix.
These processes can be done at the same time. The four-
to nine-node shape functions and their derivatives in rs-
space can be formed and then transformed into xy-space
through the inverse of Jacobian matrix.
4.2 Formulation of Shape Functions
The formulation of shape functions starts with three
basic sets of shape functions shown in Fig. 4-1.
59

1. The bilinear shape functions for four-node element.
2. The linear-quadratic shape functions for nodes
five to eight of the eight-node element.
3. The bubble shape function for node nine of
nine-node element.
These shape functions can be formulated directly from
the local coordinates of the element nodes through the
multiplication of the equations of the lines which have zero
values in the assumed displacement shapes and the scale
factor to force the shape function value to one at the node
for which the shape function is formed. The derivative of
each shape function with respect to r and s is then
evaluated from the shape function expressed in terms of r
and s.
If node nine exists, the value at node nine of shape
functions one to eight must be set to zero. The value of the
bilinear shape functions for a four-node element at node
nine is one fourth and the value of the linear-quadratic
shape functions for the five- to eight-node element at the
node nine is one half. This can be forced to zero using the
bubble shape function of the nine-node element because this
shape function has the value of one at node nine and zero at
all other nodes. Therefore the modification is the
subtraction of one fourth of the value the bubble shape
function has at node nine from the each shape function for
the corner nodes and the subtraction of one half of the

value of the bubble shape function of node nine for the
nodes five to eight, whichever exists.
If any of the center nodes on the edge of the element
(any one of nodes five to eight) exists, the bilinear shape
functions of four-node element must be modified further
because the value at center of the edge is one half in those
bilinear shape functions. This can be done by subtracting
one half of the linear-quadratic shape function for the
newly defined center node on the edge of the element from
the bilinear shape functions of the two adjacent corner
nodes. The value of any five node shape functions at the
corner node is zero. Therefore, no further consideration is
needed except for the shifting of the shape functions in the
computer implementation. These processes are shown in Fig.
4-2.
If any of the linear-quadratic shape functions of nodes
five to eight is missing, all the linear-quadratic shape
functions thereafter and the bubble shape function must be
shifted to the proper shape function number. For example, if
linear-quadratic shape function five is missing, then the
shape functions six to eight must be shifted to five through
seven and the bubble shape function must be shifted to the
node eight because all of the linear-quadratic shape
functions have been defined and numbered as shape functions
for the nodes five through eight and the bubble shape
function for the node nine.

Four Node Element
Shape Function for Corner Node
Shape Function for Edge Center Node
Shape Function for Element Center Node
Fig. 4-1 Three Basic Shape Functions

63
SF 4 = (SF 1) - (1/4) (SF 3) SF 5 = (SF 2) - (1/2) (SF 3)
Fig. 4-2 Formation of Shape Functions

4.3 The Inverse of Jacobian Matrix
While the generic displacements are expressed in terms
of rs-coordinate, the partial differential with respect to
the xy-coordinate is needed for the calculation of strain
components. Thus the inverse of the Jacobian matrix must be
calculated. This can directly be found from the chain rule
using the notation (a,b) defined as the partial differential
of function a with respect to the variable b for simplicity.
f,x = (f,r)(r,x) + (f/S) (s,x)
f,y = (f,r)(r,y) + (f,s)(s,y)
In matrix form,
f ,x
r,x
s,x
f,r
T "I
J11
j -1 '
J12
f,r
f,Y
r,y
s,y
f,s
T — 1
J21
T "I
J22
f ,s
The inverse of Jacobian matrix
But the terms in the inverse of the Jacobian matrix are not
readily available because the rs-coordinate cannot be solved
explicitly in terms of xy-coordinate. On the other hand, for
the isoparametric formulation, the geometry is interpolated
using the nodal coordinate values(constants) and the
displacement shape functions in terms of r and s. Thus the
generic coordinate x and y can be expressed in r and s

65
easily and explicit partial differentials of x and y with
respect to r and s can be performed. Therefore the Jacobian
matrix is computed and then inverted.
The Jacobian matrix is derived by the chain rule.
f,r = (f/x)( x,r) + (f,y)( y,r)
f/s = (f,x)( x,s) + (f,y)( y, s)
In matrix form,
f,r
x, r
y,r
f,x
1
l->
J12
f ,X
f ,s
x, s
y,s
f,y
J21
J22
f,y
Jacobian matrix
nn
Let E be E .
i=l
where nn = number of nodes (4 to 9).
From geometric interpolation equations,
x = E fi*x¿
y = E fi*yi
The terms in the Jacobian matrix are
li
= x, r = (s
fi*xi),r
= E
( (f1/r)
*
xi)
12
= y,r = (S
fi*yi)
= E
((fi,r)
*
Yi)
21
= X, S = (E
fi*Xi),s
= E
((fi/S)
*
xi)
22
II
*<
CO
li
M
fi*yi)/S
= s
*
Yi)

66
xi' Yi are coordinate values of the element and are
constants and therefore can be taken out of the partial
differentiation.
The inverse of two-by-two Jacobian matrix can be found
r,x = J22 / det(J)
s,x = -J12 / det(J)
r,y = -J2i / det(J)
s,y = Jji / det(J)
where deh(J) — ^ 11*^22 ^ 12^"21
4.4 Membrane Element
The formulation of the membrane element used for the
implementation follows the procedure shown on pages 115
through 118 in reference [57]. The { } symbol will be used
for the column vectors.
Nodal displacements are the nodal values of two in¬
plane translations and denoted as {u^ v^}T. The generic
displacements are defined as two translational displacements
at a point and denoted as { u v }T. By the word generic it
is meant that the displacement is measured at an arbitrary
point within an element. The generic displacements u and v
can be calculated using shape functions. The shape function
is a continuous, smooth function defined over the closed

67
element domain and is differentiable over the open domain of
the element. The shape function is also the contribution of
displacement of a node for which the shape function has been
defined to the generic displacement. Thus the generic
displacement at an arbitrary point can be found by summing
up all the contributions of all the nodes of the element.
The displacement interpolation equations are
u = 2 f¿ u¿
V = 2 £í v¡
In the isoparametric formulation the geometry is
interpolated using the same shape function assumed for the
displacement interpolation.
Therefore, the geometry interpolation is
x = E f¿
y = z fi Yi
where,
f^ = Shape function for node i.
x^, y¿ = Coordinates of node i.
u¿, Vj_ = Displacements at the node i.
u, v = Displacements at an arbitrary point within
an element

68
The three in-plane strain components for a membrane
element are
< c > = < £x ey ^xy >T
These strain components can be found through the
partial differentials of the generic displacements with
respect to xy-coordinates.
ex = u,x
ey = v,y
7xy = U/Y + v,x
Using the inverse of the Jacobian matrix, the strain
components can be evaluated.
ex
= U, X
= (u,r)(r,x) + (u,s)(s,x)
= (u,r)(J11_1) + (u,s)(J12_1)
= ((SfiUi),r) (Jn-1) + ((SfiUi)^)^-1)
= S[(fifr)(r,x) + (fi#s)(s,x)] u¿
ey
= v, y
= (v,r)(r,y) + (v,s)(s,y)
= (v,r)(J21-1) + (v,s)(J22“1)
= ((sfiV±),r)(Js!-1) + ((sfiVi),s)(J22_1)
= S[ (fifr) (r,y) + (f^sj^y)]

^xy
= u,y + v,x
= [ (u,r) (r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)]
= [((sf^) ,r) + ((Sf^i) ,s) (J22_1)]
+ [ ((ZfjVi) ,r) (Jii-1) + ((Sf^i) ,s) (J12-1) ]
= E[(fifr)(r,y) + (fi,s)(s,y)] u¿
'+ S[(firr)(r,x) + (fi,s)(s,x)] v±
New notations are introduced here to simplify the
equations. These are a^ and and defined as follows:
= (r,x)(fi,r) + (sfx)(firs) = f¿,x
*>i = (r,y)(fi#r) + (s,y)(f±,s) = fi#y
Then the strain terms above become
ex = Sa^u^ = Sf^,x u^
€y = EbiVi = sfi#y Vi
7xy = EbjUi + Sa^i = Sfi,y ui + Ef^x v±
In matrix form,
ex
ai
0
ui
£y
â– ^xy
= s
0
bi
bi
ai
vi
In symbolic form,
[ej = S[Bi][qi]

where,
ai
0
fi'X
0
0
bi
s
0
bi
ai
i
H)
H-
â–º<
fi'x
and,
[qj.]
V
i
Therefore the strain at an arbitrary point within an
element is
[e] — [B-jJ [q-jJ + [B2][q2] + •••"*" [Bg] [qg]
~ [ B1 B2 B3 B4 B5 Bg B? Bg Bg
*1
*2
*3
q4
*5
<*6
q7
^8
q9
The size of the vectors and matrix are
[«(3,1)] = [B(3,18)][q(18,l)]
In the actual calculation, this can be done by summing
up the [BjJ [qjJ over all the nodes for the given coordinates
of the point under consideration, i.e., the coordinates of
one of the integration points.

The stresses corresponding to the strains are
T
{ <7 > = { (Tx CTy TXy }
The stress-strain relationship of an isotropic material
[E]
E
11
¡21
0
0
E33
where,
E11 = E22 = E / ( 1 - p2 )
E12 = E21 = ^E / ( i ”
E33 = G
where,
E = Young's modulus
n = Poisson's ratio
G = shear mpdulus = E / ( 2*(l+/¿))
The integration over volume must be introduced for the
calculation of the element stiffness and equivalent nodal
loads of distributed loads or temperature effects. As the
thickness of the element is constant, the integration over
volume can be changed to the integration over area.
The element stiffness related to the degrees of freedom
of the node i can be calculated through the volume
T
integration of (2,3)E(3,3)(3,2).

72
[Ki]
B? E Bi dV
V
As [B] and [E] are constant about z, the integration
through the thickness from -t/2 to t/2 can be performed on z
only and yields
[Ki]
[ Bi(t)EBi] dA
A
T -
= [ Bi E Bi] dA
Ja
where,
E = tE
The size of membrane element stiffness is 18 by 18.
[K]
[ E ] [ B2 B3 ... B8 B9 ] dV
(3,3)
(3,18)
(18,3)
Equivalent nodal loads due to body forces on the
membrane element are calculated as
pb
fTb dV =
•1 r
Jv
-1 .
fTb|j|
dr ds
in which (b> = { 0 0 b2 }T or ( 0 by 0 )T or { bx 0 0 )T in

73
accordance with the direction of gravity in the coordinate
system used. The nonzero quantities bx, by, or bz represent
the body force per unit area in the direction of
application.
Equivalent loads caused by initial(temperature) strains
are
P
0
BTEe0 dV
V
r1 r1
BTE e o IJI dr ds
.-1 J-i
where,
{e0}
{ €xxO eyyO 0
{ aAT «AT 0
0 0 }T
0 0 }T
4.5 Plate Bending Element
The formulation of the plate bending element used for
the implementation has followed the procedures shown on
pages 217 through 221 in reference [57]. The { } symbol will
be used for the column vectors.
Many plate bending elements have been proposed. The
most commonly used are Kirchhoff plate elements and Mindlin
plate elements.

Kirchhoff theory is applicable to thin plates, in which
transverse shear deformation is neglected. The assumptions
made on the displacement field are
1. All the points on the midplane(z = 0) deform only
in the thickness direction as the plate deforms in
bending. Thus there is no stretching of midplane.
2. A material line that is straight and normal to the
midplane before loading is to remain straight and
normal to the midplane after loading. Thus there is
no transverse shear deformation (change in angle
from the normal angle).
3. All the points not on the midplane have displacement
components u and v only in the x and y direction,
respectively. Thus there is no thickness change
through the deformation.
Strain energy in the Kirchhoff plate is determined entirely
by in-plane strains ex, ey, and 7Xy which can be determined
by the displacement field w(x,y) in the thickness direction.
The interelement continuity of boundary-normal slopes is not
preserved through any form of constraint.
Mindlin theory considers bending deformation and
transverse shear deformation. Therefore, this theory can be
used to analyze thick plates as well as thin plates. When
this theory is used for thin plates, however, they may be
less accurate than Kirchhoff theory because of transverse
shear deformation. The assumptions made on the displacement
field are
1. A material line that is straight and normal to the
midplane before loading is to remain straight but
not necessarily normal to the midplane after

loading. Thus transverse shear deformation (change
in angle from normal angle) is allowed.
2. The motion of a point on the midplane is not
governed by the slopes (w,x) and (w,y) as in
Kirchhoff theory. Rather its motion depends on
rotations 0X and 0y of the lines that were normal to
the midplane of tire undeformed plate. Thus 0X and 0y
are independent of the lateral displacement w, i.e.,
they are not equal to (w,x) or (w,y).
It is noted that if the thin plate limit is approached, 7XZ
= 7yZ = 0 because there is no transverse shear deformation.
In this case the angles 0X and 0y can be equated to the
(w,x) and (w,y) numerically but the second assumption still
holds.
The stiffness matrix of a Mindlin plate element is
composed of a bending stiffness [kb] and a transverse shear
stiffness [ks]. [kb] is associated with in-plane strains €x,
Cy, and 7Xy. [ks] is associated with transverse shear
strains 7XZ and 7yZ. As these two groups of strains are
uncoupled, i.e., one group of the strains do not produce the
other group of strains, the element stiffness can be shown
as [82]
[k]
T
( BbEBb ) dA +
(BgEBg) dA
because BbEBs = BgEBb = 0 from uncoupling (corresponding E =
0). Each integration point used for the calculation of [kg]
places two constraints to a Mindlin plate element,
associated with two transverse shear strains 7yZ and 7ZX.
If

too many integration points are used, there will be too many
constraints in transverse shear terms, resulting in locking.
Therefore, a reduced or selective integration can prevent
shear locking. Or, the transverse shear deformation can be
redefined to avoid such locking.
For example, a bilinear Mindlin plate element responds
properly to pure bending with either reduced or selective
integration. But full two-by-two integration is used for
pure bending, shear strains appear at the Gauss points as
shown in Fig 4-3. As the element becomes thin, its stiffness
is due almost entirely to transverse shear. Thus, if fully
integrated, a bilinear Mindlin plate element exhibits almost
no bending deformations, i.e., the mesh "locks" against
bending deformations.
Nodal displacements for the plate bending consist of
one out-of-plane translation and two out-of-plane rotations
m ,
and are denoted as { w^ 6x¿ 0y¿ } . The rotations are chosen
independently of the transverse displacement and are not
related to it by differentiation. Thus the transverse shear
strains 7XZ and 7yZ are considered in the formulation
resulting in five strain components. The generic
displacements are defined as three translational
displacements and denoted as { u v w }T. By the word generic
it is meant that the displacement is measured at an
arbitrary point within an element. These generic
displacements are different quantities from the nodal

77
displacements and therefore must be related to the nodal
displacements.
The generic displacements u and v can be calculated as
functions of the generic out-of-plane rotations using the
small strain(rotation) assumption. The relationship between
generic displacements and rotation is shown in Fig 4-4.
u = z0y
v = —z*x
The generic displacements 6X and 6y can be found using the
assumed displacement shape functions and the corresponding
nodal displacements 6X^ and 0y¿.
The generic displacement w does not need any conversion
because it corresponds to the nodal displacement w¿.
In the isoparametric formulation the geometry is
interpolated using the same shape function assumed for the
displacement interpolation.
The displacement interpolation is
*x = 2 fi *xi
6y — Z f£ ^yi
W = Z f^ W¿
Similarly, the geometric interpolation is
x = Z f^ x^
y = s fi yi

Zero Shear
Strain
One Point Gauss Integration
Two Point Gauss Integration
Fig. 4-3 Shear Strains at Gauss Point (s)

Positive small rotational angle about y-axis gives
positive generic displacement in x-direction ( u ).
Shown is xz-plane.
Positive small rotational angle about x-axis gives
negative generic displacement in y-direction ( v ).
Shown is yz-plane.
Fig. 4-4 Displacements due to Rotations

80
where,
= shape function for node i
x^, = coordinates of node i
Therefore,
c
II
N
II
Z 2
fi
*yi
v = -zex =
-Z 2
fi
*xi
w =
2
fi
wi
The five strain components for plate bending element
are { ex ey 7Xy yxz 7yz }T. These strain components can be
found through the partial differentials of the generic
displacements with respect to xy-coordinates.
ex = U,X
ey = v,y
7Xy = u/Y + V,X
7XZ = u/z + W,X
7y 2 = v,z + w,y
Using the inverse of the Jacobian matrix found, the
strain components can be evaluated.
ex
= U , X = (Z0y) ,X
= (u,r)(r,x) + (u,s)(s,x)
= (u,r)(Jii-1) + (u,s)(J12_1)

Tm [(x's) (s'Tj) + (x'a)(a'Tj)]2 + =
t (x_ZTr) (^(s'Tj^)) + (x_I1:r) (tav(j/T5s) ) ] + =
[(T.ZIr)(s'(Wjs)) + (T_TIr) U' (Wjs)) ] + =
[(x's)(s'm) + (x'j)(a'w)] + =
x'm + = x'n. + z'(^0z) =
(x'w) + (z'n) = zx¿
xx0 [(x's)(s'Tj) + (x'a) ]s z-
[(A's)(s'Tj) + (A'j) (jc'Tj) ]s z =
nT_ZTr) (s' (T^Tjsz-)) + (T_XIr) U' (fVjsz-)) ] +
[ (x_Z2r) (S/(T^^TjSz)) + (x_TZr) U' ) ] =
[ (x's) (s'a) + (x'j) (j'a) ] + [(A's)(s'n) + (A'j) (a'n)] =
x‘(x0z~) + A1(^0z) = x'a + A'n =
Axl
Tx0 [(A's)(s'Tj) + (A'a)(j'Tj)]z z- =
(x_32r)(S/(TX^Tj^z-)) + (x_Tzf) (â– *'(TX0TJSZ-)) =
(x_zzr)(s'A) + (x_xZr)(a'A) =
(A's) (s'a) + (A'j) (j'a) =
A' (X0Z~) = A'a =
[(x's)(s'Tj) + (x'j)(a/Xj)3s z =
(x_ZIr) (s# (TAeT5sz)) + (x_TIr) U' (TAffT5sz)) =
18

82
7yz = (V/Z) + (W,y)
= (-Z6X),Z + w,y = (-0X) + w,y
= (-Ef^i) + [(w,r)(r,y) + (w,s)(s,y)]
= + [((sf^) ,r) (Jsi"1) +((Sfiwi) ,s) (J22-1)]
= (-sfi^xi) + [((Sfi^rjwi) (J2-L-1) +((sfi#s)wi) (J22_1) ]
= (-Sf^xi) + S[ (f±/r) (r,y) +(fi,s)(s,y)] Wi
New notations are introduced here to simplify the
equations. These are a^ and and defined as follows:
ai = (rrx)(fi#r) + (s,x)(fifs) = fi#x
*>i = (rjyHf^r) + (s^Hf^s) = fi/y
Then the strain terms above become
£x =
Z
2ai^yi
£y =
-z
2bi*xi
7xy =
z
Sbf^yi - z Sa^^xi
7xz =
Sff^yi + EajW^
N
II
-
Eff^xi + SbjW^
In matrix form,
ex
ey
7xy
= E
7xz
lyz
0
0
0
0
0
-f
i
w
i
?xi
Vi
In symbolic form,
[e] = E[BjJ [q-jj

83
where,
0
0
za^
0
- zb^
0
0
- za^
zbi
ai
0
fi
bi
- fi
0
or,
0
0
zf¿,x
0
“ zfA,y
0
0
- zfjL,x
zfi,Y
fi,x
0
fi
fi'Y
" fi
0
and,
[qjj =
wi
*xi
Vi
Therefore the strain at an arbitrary point within
element is
[«] = [B-LHq-L] + [B2][q2] + ... + [Bg] [q9]
“ [ B1 B2 B3 B4 B5 Bg B7 Bg Bg
51
52
^3
q4
55
56
57
58
59
an

The size of the vectors and matrix are
[e(5,l)] = [B(5,27)][q(27,l)]
In the actual calculation, this can be done by summing
up the [BjJ [q¿] over all the nodes for the given coordinates
of the point under consideration, i.e., the coordinates of
one integration point.
The stresses corresponding to the strains are
{ CT } = { CTX (7y Tx y Txz TyZ } ^
The stress-strain relationship of an isotropic material
is
0 0
0 0
0 0
E44 0
0 E55
where,
E11 = E22 = E / ( 1 - n2 )
E^2 = ^21 = ^E / ( 1 - ¡P")
E33 - G
E44 = E55 = G / 1.2
where,
/x = Poisson's ratio
G = shear modulus = E / ( 2*(l+/x))
[E] =
0
0
0
J11
J21
J12
322
0
0
0
0
E-
0'
0
J33

The form factor 1.2 for the E44 and E55 terms is
provided to account for the parabolic distribution of the
transverse shear stress rzx over a rectangular section. This
form factor 1.2 can be shown from the difference in
deflections of a cantilever beam at its free end [58].
Let a beam have a rectangular cross section of
dimensions b by t with a length of L. If P is the transverse
shear force, then the parabolic distribution of the
transverse shear stress rzx is
tzx = (3P/2bt3)(t2 - 4z2)
where z = 0 at the neutral axis.
Then the transverse shear strain energy from the parabolic
distribution can be calculated by
U
s
(1/2)
V
(rzxVG) dV
(1/2)
[((3P/2bt3)(t2
[(1/2)(3P/2bt3)2]/G
- 4z2))2 / G] dV
(t2 - 4z2)2 dAdz
= (area)[(1/2)(3P/2bt3)2
= bL[(1/2)(3P/2bt3)2]/G
= 1.2(P2L/btG)/2
]/G
(t2 - 4z2)2
(t2 - 4z2)2 dz
dz

While the transverse shear strain energy from the constant
distribution is
86
Us = (1/2)
(rzxVG) dV
V
= (1/2)
((P/bt)2 /G) dV
= [(1/2)(P/bt)2/G](btL)
(P2L/btG)/2
This result suggests the view that a uniform stress
P/bt acts over a modified area bt/1.2, so that the same Us
results. Therefore the deflection at the free end of a
cantilever beam with parabolically distributed transverse
shear stress will be 1.2 times that with constantly
distributed transverse shear stress.
The Mindlin plate is generalized from the Mindlin beam.
Thus the higher transverse shear stiffness from the
assumption of constant transverse shear stress has been
reduced by dividing the corresponding elastic constants by
the factor 1.2 for flat plate element. The reduced stiffness
will produce the more flexible response in shear that is
expected from the actual parabolic distribution.
The integration over volume must be introduced for the
calculation of the element stiffness and equivalent nodal
loads of distributed loads or temperature effects. As the
thickness of the element is constant, the integration over

volume can be changed to the integration over area. This can
be accomplished through the partition of the strain-nodal
displacement matrix [B¿] as follows:
87
0
0
zfi,x
0
- zf±fy
0
0
- zfitx
zfj.,y
firX
0
fi
fi'Y
- fi
0
The submatrices are named as follows:
BiA
1
<
•H
1 PQ
N
BiB
BiB
The element stiffness can be calculated
T
volume integration of (3,5)E(5,5)B^(5,3).
matrix is to be partitioned as follows:
[E]
r-
E11 E12 0
0 0
E21 E22 0
0 0
o o e33
0 0
0 0 0
E44 0
0 0 0
0 E55
The submatrices are named as follows:
E
0
E
B
through the
Thus the [E]
0

88
Then the stiffness of the element is
[K±]
T
Bi E Bjl dV
zBiA I
BiB *
dV
[z2 Ba Ea Ba] + [Bg Eg Bg]
As [B] and [E] are constant about z, the integration
through the thickness from -t/2 to t/2 can be performed on z
only and yields
[K±]
where,
[ SA(t3/12)EA5A + Bg(t)EgBg] dA
C baeaba + bbebbb] dA
EA _ (t3/12)EA and Eg = tEg
Then [KjJ can be rewritten as matrix equation as
follows:
-T T
ea
0
r ®iA i
[Kj.] =
[ BiA BiB ^
_
dA
•
0
eb
L BiB J
-T ,
B^E B¿ dA

The size of plate element stiffness will be 27 by 27.
[K] =
[ E ]
(5,5)
[ B,
B2 B3
B8 B9
] dV
(5,27)
(27,5)
The strain-nodal displacement matrix from which
the constant thickness is taken out is defined as [B^]
[B±] =
0
0
0
fi,x
fi 'Y
- fi,y
- fifX
- f;
fi,x
:i,y
fi
0
= [BiA]
L BiB j
Equivalent nodal loads due to body forces on the plate
calculated as
•
*1 |*
fTb dV
V
-1 •
fTb|j| dr ds
-1
in which (b) = { 0 0 bz }T or { 0 by 0 }T or { bx 0 0
T • . .
} m accordance with the direction of the gravity m the
coordinate system used. The nonzero quantities bx, by, or b
represent the body force per unit area in the direction of
application.

Equivalent loads caused by initial strains are
P
0
BTEe0 dV
V
r1 r1 -
5tE¿0|j| dr ds
-1 J-l
where,
T
= { ¿xxO ^yyO ^xyO 0 0 }
= { a AT/2 aAT/2 0 0 0 }T
The stresses can be calculated from the equation
M = [E][6]
The corresponding generalized stresses, if desired, may
be computed from
M = { Mxx Myy Mxy Qx Qy }T = Í ( 5 q - 0 )
It is noted that the generalized stresses are actually
moment and shear forces applied per unit length of the edge
of the plate element. Therefore these can also be turned
into common stresses using the formulation for the bending
stress calculation. The moment of inertia for the unit
length of the plate is t3 / 12. Then the in-plane stress at
a point along the thickness can be calculated as
a = Mz / I = M(t/2) / (t3/12) = 6M / t2

91
The transverse shear stresses can be found as
r = Q / t
But this may be multiplied by a factor of 1.5 to get the
maximum shear stress at a point on a neutral surface because
the transverse shear stresses show parabolic distribution
while the calculated stresses are average stresses coming
from the assumption of a constant transverse strain along
the element z axis.

CHAPTER 5
NONLINEAR SHELL ELEMENT
5.1 Introduction
The nonlinearities included in the formulation of the
Mindlin flat shell element are those due to large
displacements and those due to initial stress
effects(geometric nonlinearity). The large displacement
effects are caused by finite transverse displacements. These
effects are taken into account by coupling transverse
displacement and membrane displacements. The initial stress
effects are caused by the stresses at the start of each
iteration. These stresses change the element stiffness for
the current iteration. These effects are evaluated directly
from the stresses at the start of each iteration and are
included in the element stiffness formation.
The total Lagrangian formulation is used. If the
updated Lagrangian formulation is used, the element
coordinate system cannot be easily formed for the next
iteration because the deformed shell is not usually planar
[26], The symbol {} is used for a column matrix (a vector)
and the symbol [] is used for a matrix of multiple columns
and rows throughout the chapter.
92

93
5.2 Element Formulation
The generic displacements of Mindlin type shell element
are translational displacements {u v w)T and denoted as {U}.
The displacements and rotations at a point on the midplane
are (uQ vQ wQ 0X 0y)T and denoted as {UQ}. The generic
displacements can be expressed in terms of the midplane
displacements and z as
u = u0(x,y) + z0y(x,y)
v = vQ(x,y) - z0x(x,y) (5.1)
w = wQ(x,y)
The linearized incremental strain from Eq. (2.17) is
eij = * ( ui,j + uj,i + Sc,i uk,j + tuk,j uk,i>
(5.2)
This equation can be written out for the strain terms to be
used for shell element using the generic displacements (u,
v, w)T.
exx
X
d
ii
+
u,x
U'X
+
tv'X
V'X
+
tw'X
W,x
eyy
= u,y
+
^/y
U,y
+
tv'y
v,y
+
tw
w / y
W,y
exy
1
2
(
U,y
+
V'X
+
U,y
+
tv'X V'
y
+
W,y
+
u/x tu'y
+
v'x tv'
y
+
W,x
)
exz =
1
2
(
u»z
+
W'X
+
Vjc u,z
+
tv'X V'
z
+
tw'x W'Z
+
U»x tu/z
+
V'X tv/
z
+
w'x Vz
)
eyz =
1
2
(
V'Z
+
W,y
+
Vy U/Z
+
^ty vf
z
+
t^,y W,z
+
u'y tu/Z
+
V,y fcV,
z
+
w/y tw/Z
)
(5.
3)

The derivatives of inplane displacements u and v with
respect to x, y, and z are assumed to be small and thus the
second order terms of these quantities can be ignored
through von Karman assumption from Eqs. (2.44) [20, 21].
Furthermore the transverse displacement w is independent of
z for the shell element which means that w,z is zero. Then
Eqs. (5.3) can be reduced to
exx = u'x + tw'x W'X
d
!i
>i
0)
y
+ fcw,,
>i
¡s
p — í
«Xy 2
(
>i
d
+
V,x + W ,y + W,X )
P s 1
exz 2
(
u,z
+
W,x )
p — 1
eyz 2
(
V'Z
+
>1
¡5
(5.4)
The incremental Green's strains, sometimes called
engineering strains, can then be shown as
£xx
exx
= U'X
+
W'X W'X
>1
>1
II
(D
*<
= u,y
+
W,y
7xy
= 2exy
>i
d
II
+
V,x + W,y + W,X
7xz
= 2exz
= U/Z
+
W,x
7yz
= 2eyz
= V'Z
+
w,y
It is noted that the linearized nonlinear strains are left
only for inplane strain terms.
By substituting Eqs. (5.1) into Eqs. (5.5), the Green's
strain can be expressed in terms of midplane displacements.

95
£xx
=
U0'X
+ Ziy/X + W0'X W0'X
€yy
=
c
0
*<
" WQ / y Wq f y
7xy
=
uo'y
+ Zíy fy + VQ,X - Z0X
+
tw
wo'
x wo'y + wo'x wo'y
7xz
=
6y +
wG / x
fyz = ~9x + wo'y
This can be simplified as
{ep}
Z{eb}
{*1}
{6} =
{0}
+
{es}
{ f2 }
where,
{e} = {exx €yy 7Xy ^xz ^yz)
= Incremental strains for shell element.
{el} = {£XX ^yy 7Xy }
= Incremental inplane strains.
{£2} — Í7xz 7yz)
= Incremental transverse shear strains.
uo'x
+
wo'x wo'x
vo'y
+
Wqfy Wq/y
uo'y
+
vo'x + two'x wo'y + wo'x two'y
= Linearized incremental inplane strains.
(0) = {0 0}T
= Zero vector.
— {^y/x ~8X'y (^y/y "" 8X'x) ^
= Linear bending strains.
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)

96
{es} = {(WQ/X + 0y) (Wqty ~ 6x)}T (5.13)
= Linear shear strains.
If Piola-Kirchhoff II (PK-II) stresses are denoted as
{S}, the internal virtual work due to the virtual strain
5{e} corresponding to the virtual displacements 5{q} can be
calculated using the conjugate Green's strains as
SVIa
S{e)T{S)dV
' t/2
exsx+5 eysy+57Xysxy+5'irxzsxz+iTryzsyz)dzdA
(5.14)
where,
{S) = (Sx sy Sxy sX2 Syz)T
= PK-II stresses.
The integrand of Eg. (5.14) can be further expanded as
Integrand = [ S( ep(l) + zeb(l) )SX +
í( ep(2) + Zeb(2) )Sy +
S( ep(3) + Zeb(3) )Sxy +
5eS(l)Sxz + 5eS(2)Syz ]dzdA
If the integration over thickness is performed analytically,
this can be rearranged as
Integrand = [ 5{ep)T{SSp) + 5{eb)T{SSb)
+ S(es}T{SSs} ]dA
[5{ee}T{SS} ]dA

where,
{ce} = { = Incremental shell element strains grouped
into inplane, bending, and shear strains.
{SS} = { (SSp}T {SSb}T {SSs}T }T
= Generalized stresses arranged for {ee}.
t/2
{SSp} =
r -]
* t/2
Nx
SX
dz =
N„
ss
s„
Nxy
-t/2
sy
bxy
«
•
{SSb} =
{SI} = {Sx Sy Sxy}
T
-t/2
{Sl}dz
(5.15)
CM
M
llX.
SX
M
y
M
1Axy
-t/2
sy
sxy
zdz =
•
t/2
-t/2
{Sl}zdz
(5.16)
* t/2
Qx
sxz
(SSs) =
[ qy J
•
-t/2
syz
dz =
t/2
-t/2
{S2}dz
(5.17)
= Inplane Piola-Kirchhoff II stresses.
“ = Transverse shear PK-II stresses.
These generalized stresses (stress resultants along the
element thickness per unit length) can be further expanded
incorporating the stress-strain relationship between PK-II
stress and the conjugate Green's strain.

98
{S} = [E]{£>
where the constitutive matrix [E] can be subdivided as
[E (5,5) ] =
[El] =
[E2 ]
(SSp)
l-/x‘
E
[El](3,3) [0 ] (3,2)
[0 ] (2,3) [E2](2,2)
1 n 0
H 1 0
0 0 (1-^/2
(1-J0/2 0
0 (l-n)/2
t/2
(Sl)dz =
t/2
[El]{eljdz
-t/2
t/2
-t/2
= Integl + Integ2
-t/2
[El]{£p + Zfbjdz
where,
Integl =
t/2
[El](ep)dz
Integ2 =
[El]{ Z£b }dz
(i
-t/2
't/2
-t/2
For a single layer element, integ2 is always zero.
(SSb) =
' t/2
-t/2
t/2
-t/2
z{Sl}dz =
' t/2
-t/2
z[El]{£l}dz
[El](Z£p + z^fbjdz
.18)
(5.19)

99
{SSs}
t/2
-t/2
' t/2
{S2}dz =
t/2
[E2] { e2}dz
-t/2
[E2]{es}dz
(5.20)
“t/2
Eqs. (5.18),(5.19) and (5.20) can be put together to form
the generalized "stress-strain" relationship of a shell
element as
(SS)(8,1) = [D](8,8){££>(8,1)
where,
[D]
CDiiia^S)][D12(3# 3)3 CO (3,2)]
[D21(3,3)][D22(3,3)][0 (3,2)]
[0 (2,3)][0 (2,3)][D33(2,2)]
(5.21)
[Du] =
't/2
-t/2
[El]dz
tD123 “ [°21^
’t/2
z[El]dz
-t/2
[d22]
[D33]
t/2
-t/2
't/2
-t/2
z2[El]dz
[E2]dz
It is noted that [D12] and [D21] are zero for a single layer
element because it is the integration of an odd function
over the open domain (-t/2,t/2). For a multi-layer system,
these must be kept as is because material properties of
layers change as the element deforms.

o o o o o
100
5.3 Finite Element Discretization
5.3.1 Linearized Incremental Strain-Displacement
Relationship
By rearranging Eqs. (5.11) through (5.13), the strain-
inidplane displacement relationship is
(ee)
eb
eS
U0'X
+
(twO'X
>1
0
>
+
(two'y
uo'y
+
vo' X +
^y'x
“flx'y
9Y'Y
-
6X'X
X
0
>
+
eY
wo»y
-
0*
(wo'x)
(wo'y>
(two'x)(wo'y)
+ (^o^) (wQ,x)
d/dx
0
d/dy
0 (hr
d/dy (hr
d/dx (rw
O'X
o'y
O'X
0 0
0 0
0 0
0 d/dx
0 d/dy
)(d/dx)
)(d/dy)
) (d/dyi + fS^y
0
0
)(d/dx) 0
0
-d/dy
-d/dx
0
-1
0
0
0
d/dx
0
d/dy
1
0
u.
w
[d](8,5) [U0](5,l)
(5.22)

101
5.3.2 Generic Displacement-Nodal Displacement Relationship
The generic displacements at midsurface {U0} can be
interpolated using the shape (interpolation, displacement)
functions and the nodal displacements at midsurface (q).
(U0>
uo
!iui
vo
n
fivi
wo
= s
fiwi
«X
i=l
!i*xi
[ *Y J
f i^yi
flul
+
f2u2
+ •
• +
£nun
!lvl
+
!2v2
+ •
• +
?nvn
!iwi
+
f 2W2
+ *
• +
fnwn
£l*xl
+
1l29x2
+ •
• 4*
^n^xn
f l^yl
+
t2ey2
+ •
• +
fn*yn
fx 0 0 0 0 f2 0 0
0 fx 0 0 0 0 f2 o
0 0 fx 0 0 0 0 f2
000 f± 0000
0000 fx 000
fn 0 0 0 0
0 fn 0 0 0
0 fn 0 0
0 f„ 0
0
0
0
0
0
0n f
n
Ui
W,
'xl
'yi
u
w-
'x2
?y2
u
w.
n
n
n
xn
yn

102
= [ f![15] f2[I5] ... fn[I5]
qi
<*2
q
n
= [[%] [N2] ... [Nn3 3 {q}
{UQ} = [N]{q}
where,
n = Number of element nodes.
[15]
1 0 0 0 0
0 10 0 0
0 0 10 0
0 0 0 1 0
0 0 0 0 1
[N] = [ ^[15] f2[I5] ... fn[I5] ]
= C [% ] [N2 ] ... [Nn] ]
{gi} = {uL Vi Wi exi eyi}T
(q) = { {qi)T (q2}T ••• {qn>T >T
5.3.3 Strain-Nodal Displacement Relationship
If Eq. (5.24) is substituted into Eq. (5.22), the
incremental strain is
{ee) = [d]{U0)
= [d] [N](q)
= [Blz] (q>
(5.23)
(5.24)
(5.25)
(5.26)

103
where,
[Blz] = [d][N] (5.27)
= Linearized strain-nodal displacement matrix.
5.3.4 Evaluation of Linearized Strain-Nodal Displacement
Matrix
Each term of the [BLZ] matrix is calculated using Eq.
(5.27) .
[Blz] = [d] (8,5) [N] (5, (5n))
= [d] [ [N-l] (5,5) [N2](5,5) ... [Nn](5,5) ]
Let [B-jJ = [dHNjJ, then
Cblz^ “ tBi b2 ••• Bn^
where,
[B±] = [dJCNi]
d/dx 0 (íw0,x)(d/dx)
0 d/dy (rw0,y)(d/dy)
d/dy d/dx (S^,*) (d/dy)
0 0 0
0 0 0
0 0 0
0 0 d/dx
0 0 d/dy
0 0
0 0
w0,v)(d/dx) 0 0
2 0 d/dx
-d/dy 0
-d/dx d/dy
0 1
-1 0
fi 0 0 0 0
0 0 0 0
0 0 fj, 0 0
0 0 0 f¿ 0
0 0 0 0
X

104
(fi'
x)
0
0
0
0
(fi'
y>
(*• i'y)
(fi'x>
tw°fy (fi'y) t
(Si'y) (fi/^ + i^o^x) (fi'y)
0
0
0
0
0
0
0
0
( f i / x)
0
0
0
0
0
0
0
(fi/y)
0
0
(!i'x>
0
fi
0
0
(fi/y)
-fi
0
being denoted as
[BlPi](3,2)
(0] (3,2)
[0] (2,2)
[Bn, )(3,3)
[BlÉil(3,3)
[BlS-jJ (2,3)
(5.28)
and decomposed into two parts for later use in element
stiffness calculation.
[BlPi
] (3,2)
[0]
(3,3)
[Bl] =
[0]
(3,2)
[Blbi
] (3,3)
(5
.29)
[0]
(2,2)
[BlSi
3(2,3)
[0]
(3,2)
[Bn±
] (3,3)
[Bn] =
[0]
(3,2)
[0]
(3,3)
(5
30)
[0]
(2,2)
[0]
(2,3)
5.3.5 Evaluation of Nonlinear Strain-Nodal Displacement
Matrix
From Eq. (2.17), the nonlinear incremental strain components
ij * (uk,i uk,j)
are

105
If Eq. (5.1) is substituted into Eq. (5.30), then
^XX “ *
(U/X
u, X
+
V'X
V'X
+
W,x
W,x
r,yY - 1
(u,y
U,y
+
V/y
v,y
+
W,y
W, y
»7xy = |
(u f X
U/y
+
V'X
v,y
+
w,x
W,y
^xz = *
u,z
+
V'X
V'Z
+
W,x
W,z
»?yZ = 5
(U,y
U,z
+
v,y
V'Z
+
W'Z
W, z
This can be simplified using the same assumptions for
linearized strains.
’'xx =
1
2
w,x
w,x
-3
II
1
2
W,y
W,y
^xy =
1
2
W,x
W,y
(5.32)
ixz = 0
riyZ = 0
And only non-zero terms are contained for corresponding
Green's strain and denoted as (n).
nxx ^xx *
W,x
w,x
^yy »?yy — 2
W,y
W,y
(5.33)
nxy = 2r?xy =
W,x
W,y
If Eq. (5.1) is substituted into Eq. (5.33), then [49]
(n)
I(wQ,x
i(wO'y
(wo'x
) (wo
'Y
)
wO' X
0
W0'X
_ 1
— 1
0
Wq / y
wo'y
wo'x
Wq / y
= HA]{i) (5.34)

106
As [A] and {0} are linear functions of {q>, i.e.,
linear functions of (w,x) and (w,y), the strain is quadratic
in nodal displacements. But the (w,x), (w,y) values from the
previous iteration are used in [A] as approximations to the
true values. The strain is linearized in this manner and
used in the evaluation of element stiffness matrix for the
iterative solution of nonlinear equilibrium equation.
To apply the principle of virtual work, it is desirable to
express nonlinear strain in terms of nodal displacements
{q}-
. {n} = \ [A]{*> = f({q})
and the displacement gradient {0} can be written in terms of
{q}-
n
[2
i=l
(fjWi)]
'X
n
[2 (f^) ] ,y
i=l
(?l'x>wl + if2'x)w2
(fl,y)wl + (f2,y)W2
+
+
+ J?n'x!wn
+ (fn,y)wn

107
0 0 0 0 0 0 f2»jj
0 0 flfy 00 00 f2,y
0 0 fn,x 0 0
0 0 fn,y 0 0
Ui
w
u
1
xl
yi
w-
*x2
9y2
u
w.
n
rn
n
xn
yn
= [ [Gi] [G2]
[Gn] ]
*1
*2
q
n
= [G]{q}
where,
(5.35)
[GiJ
0 0 (fi#x) 0 0
0 0 (fi/y) 0 0
[G] = ( [Gx] [G2]
tGn] ]

108
5.3.6 Discretization
The word "discretization" means that the continuous
displacement field is approximated using displacements at
discretized nodal points.
For the total Lagrangian formulation, incremental
iterative equilibrium equation from Eq. (2.25) is
oCijkl Qekl
eii
o
3dV 4
°dV
= EVW
°dV
This can be shown in a matrix form as follows. The term
"linear" means the first order differential of displacements
with respect to coordinate variables while the term
"nonlinear" means the second order differential of
displacements with respect to coordinate variables. For
example, if (u,x) is defined as au / ax and (u,xx) is
defined as 32u / ax2, (u,x) is "linear" while (u,xx) is
"nonlinear" in reference to strain terms.
The linearized incremental strain (e) is related to
incremental nodal displacements {q} through linearized
strain-displacement matrix [BLZ] from Eq. (5.27).
(e> = [BLZ]{q)

109
The variation of linearized incremental strain is then
5{e} = i[BLZ]{q} + [BLZ]i{q} = [BLZ]5{q> (5.36)
because [BLZ] is constant about the unknown incremental
displacements and the variation thereof is zero, i.e.,
S[Bl2] is zero.
The nonlinear incremental strain {n} can be shown as
the multiplication of two matrices [A] and {£}, which
contains only linear terms. From Eq. (5.34),
(n) = | [A]{0>
The virtual variation in nonlinear strain terms is
s(n> = sl[A]{0}
= l(fi[A)){i) + I[A](5{0 } )
= HA](S{0}) + i[A](i{i>)
= (A] (5{^ >) (5.37)
because (5[A]){0) is equal to [A](5{^}) as shown below.
5W0'X
0
wo'x
0
íw0,y
wo'y
*wo'y
5W0'X
[A]6(0} =
w,
0
w,
O'X
o'y
w,
o'y
O'X
(*w0,x)
(5wo'y
Thus,
(£ [A]){8} = [A](8{8})
*(w0,x)(wo'x)
5(wo'y^ (wo'y)
*(wo'x> (wo'y) + iwo'x)5(wo'y)

The gradient of displacements {0} is related to incremental
nodal displacements through [G] matrix. From Eq. (5.35),
{0} = [G] {q)
and the variation thereof is
5(0} = 5[G]{q) + [G]5{q) = (G]5{q) (5.38)
because [G] is constant about the unknown incremental
displacements and the variation thereof is zero, i.e., 5[G]
is zero.
Now the incremental iterative equilibrium equation can
be put into a matrix equation. It is noted that the
engineering strains, (e) and (n), are used in the places of
ekl' eij and ^ij corresponding constitutive matrices.
cijkl ekl 5 eij = 5ie}T[cHe}
= ([BLZ]5{q))T[C](CBlz](q) )
= «(q}T[BLZ]T[C][BLZ]{q}
= 6(q}T[K1]{q}
sij 5 *lij = S{n}T{S)
= ([A]5{0})T{S)
= 5<0}T[A]T{S}
= ([G]5{q})T[A]T(S)
= 5{q)T[G]T(S]{0)
= *{q>T[G]T[S][G]{q}
= 5{q}T[K2]{q)
The relationship, [A]T{S) = [S]{0> [49], is simple
mathematical equivalence by rearranging the elements of the
matrices in different format to relate the nonlinear strain

Ill
{n} to incremental displacement {q>. It is noted that [S] is
a multi-column and multi-row matrix and {S} is a column
matrix (a vector). This will be discussed in 5.4. The
matrices [K-jJ and [K2] are newly defined as
[*l] = [BLZ]T[C][BLZ]
[K2] = [G]T[S][G]
Sij 8 e^j = 5{e}^{S}
= «{q}T[BLZ]T{S}
External virtual work due only to nodal forces is
EVW = 5{q}T{P}
Then the incremental equilibrium equation becomes
S{ q)T
[KiHq) dV + 5{q)T
[K2]{q) dV
= S{q}T{P) - í{q}T
[Blz]T{S) dV
Let
Cklz] =
[%] dV
[Kg] =
[K2] dV

112
{RI}
[Blz]T{S) dV
If the volume integration is changed to area integration
using analytical integration through thickness,
[rlz]
[Kj] dA
(5.39)
[KG]
[Kn] dA
(5.40)
{RI}
[Blz]T{SS} dA
(5.41)
where,
[Kil “ [Blz]T[D][Blz]
[Kj-j-] = [G]T[SS][G]
It is noted that stresses are in a resultant form [SS] with
the corresponding constitutive matrix [D].
Then the equilibrium equation becomes
i(q)T (( [Klz] + [Kg] ) (q) - (P) + (RI) ) = {0}
and this must be satisfied for any virtual displacements,
5{q}, meaning that ${q} cannot always be {0}, thus,
(( [Klz] + [Kg] ) {q} - {P} + {RI} ) = {0}
And finally the usual form, [K] {q} = {R}, can be obtained.
( [Klz] + [KG] ) {q} = {P} - {RI} (5.42)

113
5.4 Derivation of Element Stiffness Matrix
5.4.1 Linearized Element Stiffness
The linearized incremental element stiffness due to
linear and large displacement effects is evaluated using Eq.
(5.39).
For the efficiency in calculation, [BLZ] is divided into
[Bl] and [Bn] in Eqs. (5.29) and (5.30), then
[Blz]T[D][Blz] = [[Bl]+[Bn]]T [D] [[Bl]+[Bn]]
= [B1JT[D][B1]
+ [Bl]T[D][Bn] + [Bn]T[D][Bl] + [Bn]T[D][Bn]
[klz3
[B1]T[D][B1] dA
+
([B1]T[D][Bn]+[Bn]T[D][Bl]+[Bn]T[D][Bn]) dA
[Kl] + [Kid]
(5.43)
where,
[Kl] =
[B1]a[D][B1] dA
(5.44)
= Linear element stiffness.
[Kid] =
( [B1]T[D] [Bn] + [Bn]T[D] [B;L] + [Bn]T[D] [B„] ) dA
= [Kldl] + [Kld2] + [Kld3]
(5.45)
= Large displacement element stiffness.

114
5.4.2 Geometric Element Stiffness
The element stiffness due to initial stress effects is
calculated using Eq. (5.40). The [SS] matrix must be found
using [A] and {SS}.
From Eq. (5.34),
[A] =
wo'x 0
0 w
w
o'y
w
o'y
o'x
{*} =
WG'X
wo' y
The {SS} corresponding to nonlinear strain {n> is
{SS} = {Nx Ny Nxy}
The relationship between [SS] and {SS} is by simple
rearrangement of matrix elements.
[A]t{S} = [SS]{0}
wo'x ® wo'y
0 w0#y wQ,x
x
xy
Nx N
Nxy Ny*
w0 r X
WQf y
Thus matrix form of generalized stresses [SS] is defined as
[SS] =
Nx N
NXy N/
(5.46)

Thus the geometric element stiffness becomes
[Kg]
[G]T[SS][G] dA
(5.47
= Geometric element stiffness.
5.5 Calculation of Element Stiffness Matrix
5.5.1 Calculation of TK11
Each term of linear element stiffness is calculated
from Eq. (5.42) block by block.
From Eq. (29),
[Bli]T =
[BlPi]
T(2,3)
[0]
t(2,3)
[°] m
(2,2)
[0]
(3,3)
[BlbjJ
T(3,3)
[BlSi]T
(3,2)
[Blp-i ]
[0]
(3,2)
[0]
(3,2)
[0]
(2,2)
[Blj] =
(3,3)
[Blbj]
(3,3)
[BlSj]
(2,3)
From Eq. (5.21),
[D]
(D11(3,3)][D12(3,3)][0 (3,2)]
[D21(3,3)][D22(3,3)][0 (3,2)]
[0 (2,3)][0 (2,3)][D33(2,2)]
Thus,
[B1]T[D][B1](5,5)
[Kipp](2,2) [Klpb] (2,3)
[Klbp](3,2) [Klbs] (3,3)

116
where,
[Kipp] = [Blpi]T[D11][Blpj]
[Klpb] = [Blpi]T[D12][Blbj]
[Klbp] = [Blbi]T[D21][Blpj]
[Klbs] = [Blbi]T[D22][Blbj] + [Blsi]T[D33][BlSjJ
Note that for a single layer element, both [Klpb] and [Klbp]
are zero matrices.
5.5.2 Calculation of TKldl
Each term of nonlinear element stiffness from large
displacement effects is calculated from Eq. (5.45) block by
block.
/
5.5.2.1 Calculation of TB11—TD1TBnl for TKldll
[B1]t[D][Bn]
[0] (2,2) [Kin] (2,3)
[0] (3,2) [0] (3,3)
where,
[Kin] = [Blpi]T[D11][Bnj]
Actual calculation gives
Kln(l,1) 0 0
Kln(2,1) 0 0
[Kin] =

117
where,
Kin(1,
1)
D11
(If
1)
(fi-
X^
Bnj
(1.
â– 1)
+
D11
(I,
2)
(fi<
'x)
Bnj
(2,
1)
+
D11
(3,
3)
(fi<
-y)
Bnj
(3,
-1)
Kin(2,
1)
=
D11
(2,
1)
(fi<
ry)
Bnj
(1-
-1)
+
D11
(2,
2)
(f±,
ry)
Bnj
(2,
-1)
+
D11
(3,
3)
(fi-
rX^
Bnj
(3,
rl)
[Bnj] = [Bn] evaluated for node j
5.5.2.2 Calculation of TBnl—TD1TB11 for TKld21
[Bn]T[D][Bl]
[0] (2,2) [0](2,3)
[Knp](3,2) [Knb](3,3)
where,
[Knp] = [Bn^tD^nBlpj]
[Knb] = [Bni^tD^Hblbj]
Note that for a single layer element, [Knb] is zero matrix.
Actual calculation gives
[Knp]=
Knp(1,1) Knp(1,2)
0 0
0 0
where,
Knp(1,1) = ( D11(l,l)Bni(l,l)+D11(2,l)Bni(2,l) )(fj,x)
+ Dj^íS^Bnip,!) (fj,y)

118
Knp(l,2) = ( D11(l/2)Bni(l,l)+D11(2,2)Bni(2/l) )(fj,y)
+ D11(3,3)Bni(3/l)(fj/x)
[Bnj] = [Bn] evaluated for node j
[Knb]
0 Knb(1,2) Knb(1,3)
0 0 0
0 0 0
where,
Knb(1,2) = (D12(l/2)Bni(l,l)+D12(2,2)Bni(2,l))(-fj/y)
+ D12(3#3)Bni(3,l)("fj,x)
Knb(1,3) = (D12(l,l)Bni(l,l)+D12(2fl)Bni(2fl))(fj#x)
+ D12(3,3)Bni(3,l)(fj/y)
5.5.2.3 Calculation of fBnl—TD1TBnl for TKld31
[Bn]T[D][Bn]
[0](2,2) [0] (2,3)
[0](3,2) [Knn] (3,3)
where,
[Knn] = [Bni]T[D11][Bnj]
Actual calculation gives
[Knn]
Knn(1,1) 0
0 0
0 0
0
0
0

where,
Knn(l,1) = (D11(l/l)Bni(l,l)+D11(2/l)Bni(2,l))Bnj(l,l)
+ (D11(l,2)Bni(l,l)+D11(2,2)Bni(2,l))Bnj(2,l)
+ D11(3,3)Bni(3,l)Bnj(3,1)
5.5.3 Calculation of TKC1
Each term of nonlinear element stiffness from stress
effects is calculated from Eq. (5.47) block by block.
[Gi^tSSHGj]
0 0 0 0 0
0 0 0 0 0
= 00 Kg(3,3) 0 0
0 0 0 0 0
0 0 0 0 0
where,
Kg(3,3) = Nx(fi/x) (fjfx)
+ Nxy[(fi/x)(fj,y)+(fj,x)(fi/y)]
+ Ny(fi/y) (fj,y)
5.6. Element Stress Recovery
Stresses can be calculated from the Eqs. (5.15), (5.16)
and (5.17).
{SSp}
t/2
(Sl)dz =
-t/2
*t/2
[El]{el)dz
-t/2

*t/2
[El]{ep + Zeb}dz
-t/2
By explicit integration along thickness
nlyr
{SSp} = E [El][(ep*thk(k) +cb*thk2(k)]
k=l
(5.48)
where,
nlyr = number of layers
thk(k) = the thickness of k-th layer
thk2(k) = | (h(k+l)2 - h(k)2)
h(k) = the dimension from bottom of element to the
bottom of k-th layer
Similarly, {SSb} and {SSs} can be found as
{SSb} = z{Sl}dz = z[El]{el}dz
[El](Zep + z2eb)dz
nlyr
E [El][ep*thk2(k) +eb*thk3(k)]
k=l
(5.49)
where
thk3(k) = (1/3)(h(k+l)3 - h(k)3)
{SSs} = {S2}dz = [E2]{e 2}dz
[E2](eS)dz
nlyr
E [E2][thk(k)eS]
k=l
(5.50)

These stresses are generalized stresses. Thus the
common stresses for the determination of layer or element
state must be calculated using the definition of generalized
stresses from Eqs. (5.48), (5.49) and (5.50).
From {SSp}, i.e., (Nx Ny NXy}T, which are the
resultants of the inplane stresses for unit length of
element edges, PK-II stresses Sx, Sy, and SXy can be
calculated as
'x
= Nx / thk
S„ = Nv / thk
Sv„ = Nvv / thk
y y
'xy ” iTxy
From the (SSb), i.e., (Mx My MXy), which are the moment
resultants of the inplane stresses for the unit length of
element edges, PK-II stresses Sx, Sy and SXy can be found as
'x
Mx(thk/2) / II = Mx * S6
Sy = My(thk/2) / II = My * S6
SXy = This is a torsional moment and very complicated
in nature but can be approximated as (1/3)(1-0.63thk)thk3.
where,
thk = Element or layer thickness.
II = Moment of inertia of the unit length of element
section.
= (1)(tkh3)/12
S6 = The inverse of section modulus of the unit length
of the element section.
= 11/(thk/2) = thk2/6

And these two components from {SSp} and {SS^} must be summed
up for total stresses.
Similarly PK-II stresses Sxz, and SyZ can be calculated
from {SSs}.
sxz = Qx / thk
Syz = Qy / thk
5.7. Internal Resisting Force Recovery
Once the general stresses are obtained, internal
resisting forces at node i can be evaluated as
=
,T
[Bi]x{SS} dA
(5.51)
where,
(Pi) = {PXi, Py^ Pzi# Rxit Ry-jJ
T
{Pmi}(2,l)
{PPj_} (3,1)
Px¿, Py¿, Pz¿ = Concentrated nodal forces in x, y, z
directions.
Rx¿ = Nodal moment about x-axis.
Ry¿ = Nodal moment about y-axis.
{Pmi}= {PXi, Pyi}T
= Concentrated nodal forces from membrane
behavior.

123
(Ppi}= {Pzif Rx±l RYi}1
= Concentrated nodal forces from plate behavior.
CB±]
[BlPi](3,2) [Bni] (3,3)
[0] (3,2) [BlbjJ(3,3)
[0] (2,2) [Blsi](2,3)
(SS) = { (SSp}T (SSb}T {SSs}T }T
— { N, Ny , Njjy , , My , Mj^y , Qj£ , Qy }
Note: These generalized stresses are evaluated at the
current integration point for the numerical integration of
the internal resisting forces.
The integrand of Eq. (5.51) can be evaluated using
submatrices as
(Pmi) = [BlPi]T{SSp}
{PPi> = [Bni]T{SSp) + [Blbi]T{SSb) + [Bls^tSSs}
Actual calculation will be
(Pmi) =
[Blpi]T{SSp} =
(fi'x)
0
(fi'y)
Nx
0
(fi'y)
(fi/X)
Ny
Thus,
PXi - (fi,X)*Nx + (fi,y)*NXy
PYi = (fify)*Ny + (fi,X)*NXy
The concentrated nodal forces from plate behavior have three
components.

124
The first component is
[Bni]T{SSp> =
Bnll Bn21 Bn31
Nx
0 0 0
Nv
0 0 0
xy
Thus,
Pzp^ = Bni;L*Nx + Bn21*Ny + Bn31*Nxy
Rxp¿ = 0
RyPi = 0
where,
Bni;L = Bnp^ (1,1)
Bn21 = Bnp^(2,l)
Bn31 = Bnp¿(3,l)
Note : These are all the nonzero terms in [BlpjJ .
Pzp¿ = Component of Pz¿ from nonlinear strain terms,
Rxp¿ = Component of Rx^ from nonlinear strain terms,
RyPj. = Component of Ry^ from nonlinear strain terms,
The second component is
[Blib]T{SSb} =
0
0
0
MX
0
-(fi,y)
M
y
(fi'x)
0 J
(fi/y)
M"*
X1xy
Thus,
Pzb¿ =
0
Rxb¿ =
-(fi,y)*My -
Rybi =
(fi,x)*Mx +
(fi,y)*Mxy

125
where,
Pzb¿ = Component of Pz¿ from plate behavior.
Rxbj^ = Component of Rx¿ from plate behavior.
Rybi = Component of Ry^ from plate behavior.
The third component is
[Blis]T{SSs) =
Qx
QY
Thus,
Pzsi = (fi#x)*Qx + (fi/y)*Qy
Rxsi = -(fi)*Qy
Rys± = (fi)*Qx
where,
Pzs¿ = Component of Pz¿ from shear stresses.
Rxs^ = Component of Rx¿ from shear stresses.
Rys¿ = Component of Ry^ from shear stresses.
Therefore the concentrated nodal forces are
Pzi = Pzp¿ + Pzb¿ + Pzs¿
Rx^ = Rxp¿ + Rxb¿ + Rxs^
(fi) 0
RYi = RyPi + Ryt>i + Rys¿

CHAPTER 6
NONLINEAR SHELL ELEMENT PERFORMANCE
6.1 Introduction
The nonlinear Mindlin shell element directly derived
from the linearized incremental equilibrium equation
presented in chapter 5 has been implemented in the general
nonlinear analysis program ANSR developed at the University
of California, Berkeley [56], Linear material property is
assumed for all the test runs. Three commonly used examples
are tested. These are a cantilever beam with free end
moment, a clamped square plate with distributed load, and a
simply supported square plate with distributed load.
6.2 Large Rotation of a Cantilever
For the cantilever with free end moment shown in Fig.
6-1, the analytical solution can be found as follows.
From the geometry, the length does not change as the
beam deforms.
2jtR * { / 2n) = L
Thus,
R = L / (6.1)
126

127
From the moment-curvature relationship,
= ML / El (6.2)
From Eq. (6.2),
M = El / L (6.3)
and if Eq. (6.2) is substituted into Eq. (6.1),
R = El / M (6.4)
The free end displacements u, v can be found using
geometry.
u = L - R sin^
= L - (EI/M) sin (ML/EI) (6.5)
v = R - R cos<¿
= R ( 1 - cos )
= (EI/M)( 1 - COS (ML/EI)) (6.6)
The data used are
E = 30000 ksi
I = (1)(0.1)3/12 in4
El = 2.5 Kips-in2
L = 10 in
The analytical solution and numerical solution from
ANSR are given in Table 6-1 and plotted in Fig. 6-2 and Fig.
6-3 for the vertical displacement and horizontal
displacements. The figures show the excellent response
within moderate rotation limits, in this case a total change

in angle of n/8 radians. They also show the deviation in
horizontal and vertical displacements as the rotation
becomes large.

Width = 1 in.
Fig. 6-1 Cantilever under Free End Moment

Table 6-1 Displacements of Cantilever Beam
under Free End Moment
Load
Level
Moment
(K-in)
Z-Disp.
ANSR
Z-Disp.
ANAL
X-Disp.
ANSR
X-Disp.
ANAL
0
0.0000
0.0000
0.0000
0.0000
0.0000
1
0.0196
0.3926
0.3925
0.0205
0.0103
2
0.0393
0.7819
0.7838
0.0611
0.0411
3
0.0589
1.1660
1.1727
0.1208
0.0923
4
0.0785
1.5430
1.5579
0.1982
0.1637
5
0.0982
1.9100
1.9384
0.2919
0.2550
6
0.1178
2.2690
2.3129
0.4000
0.3660
7
0.1374
2.6160
2.6803
0.5211
0.4962
8
0.1571
2.9530
3.0396
0.6533
0.6451
9
0.1767
3.2780
3.3896
0.7952
0.8122
10
0.1963
3.5920
3.7297
0.9454
0.9968
11
0.2160
3.8960
4.0576
1.1030
1.1984
12
0.2356
4.1890
4.3737
1.2660
1.4161
13
0.2553
4.4720
4.6767
1.4340
1.6491
14
0.2749
4.7450
4.9657
1.6060
1.8967
15
0.2945
5.0100
5.2399
1.7820
2.1579
16
0.3142
5.2650
5.4987
1.9610
2.4317
17
0.3338
5.5120
5.7412
2.1410
2.7173
18
0.3534
5.7510
5.9670
2.3240
3.0135
19
0.3731
5.9330
6.1755
2.5080
3.3194
20
0.3927
6.2080
6.3662
2.6930
3.6338

Displacement (in)
4
Fig. 6-2 Vertical Free End Displacement of Cantilever Beam
under Free End Moment
131

Displacement (in)
Fig. 6-3 Horizontal Free End Displacement of Cantilever Beam
under Free End Moment
132

133
6.3 Square Plate
The second test model used is shown in Fig. 6-4. This
is a square plate under distributed loads. The boundary
conditions can either be fixed or simply supported. The data
used are
n = 0.3 = Poisson's ratio
a = 300 in = Side length
t = 3 in = Thickness
E = 30000 ksi
q = Distributed load
The analytical linear solutions for the displacement at
center of plate [59] are
w = 0.00126qa4/D for clamped square plate
w = 0.00406qa4/D for the simply supported square plate
where, D = Et3 / 12(l-/¿2) = plate stiffness
The linear analytical solution and numerical solution
from ANSR are given in Table 6-2, Table 6-3 and plotted in
Fig. 6-5 and Fig. 6-6 for clamped plate and simply supported
plate, respectively. The comparison of nonlinear responses
is given in Tables 6-4 and 6-5.
The relative effects of large displacements and initial
stresses with respect to total nonlinear effects are given
in Table 6-6 and Fig. 6-7.

134
Size = 300 in. x 300 in.
Thickness = 3 in.
E = 30000 ksi
Poisson's ratio = 0.3
Fig. 6-4 Square Plate under Distributed Loads

Table 6-2 Displacements of Square Plate with
Fixed Support under Distributed Loads
Step
No.
Wt
(pcf)
q
(psi)
Linear
(in)
ANSR
(in)
0
0
0.0000
0.0000
0.0000
1
1000
1.7361
0.2389
0.2389
2
2000
3.4722
0.4778
0.4716
3
3000
5.2083
0.7166
0.6949
4
4000
6.9444
0.9555
0.9065
5
5000
8.6806
1.1944
1.1060
6
6000
10.4167
1.4333
1.2930
7
7000
12.1528
1.6721
1.4680
8
8000
13.8889
1.9110
1.6310
9
9000
15.6250
2.1499
1.7850
10
10000
17.3611
2.3888
1.9300
11
11000
19.0972
2.6276
2.0660
12
12000
20.8333
2.8665
2.1950
13
13000
22.5694
3.1054
2.3180
14
14000
24.3056
3.3443
2.4340
15
15000
26.0417
3.5831
2.5440
16
16000
27.7778
3.8220
2.6500
17
17000
29.5139
4.0609
2.7510
18
18000
31.2500
4.2998
2.8470
19
19000
32.9861
4.5386
2.9400
20
20000
34.7222
4.7775
3.0290

Displacement (in)
5
Fig. 6-5 Center Displacement of Clamped Square Plate
under Distributed Load
136

Table 6-3 Displacements of Square Plate with
Simple Support under Distributed Loads
Step
No.
Wt
(pcf)
q
(psi)
Linear
(in)
ANSR
(in)
0
0
0.0000
0.0000
0.0000
1
1000
1.7361
0.7697
0.6618
2
2000
3.4722
1.5394
1.1340
3
3000
5.2083
2.3091
1.4940
4
4000
6.9444
3.0788
1.7850
5
5000
8.6806
3.8485
2.0290
6
6000
10.4167
4.6183
2.2400
7
7000
12.1528
5.3880
2.4260
8
8000
13.8889
6.1577
2.5930
9
9000
15.6250
6.9274
2.7450
10
10000
17.3611
7.6971
2.8850
11
11000
19.0972
8.4668
3.0140
12
12000
20.8333
9.2365
3.1340
13
13000
22.5694
10.0062
3.2470
14
14000
24.3056
10.7759
3.3540
15
15000
26.0417
11.5456
3.4540
16
16000
27.7778
12.3153
3.5500
17
17000
29.5139
13.0850
3.6410
18
18000
31.2500
13.8548
3.7290
19
19000
32.9861
14.6245
3.8120
20
20000
34.7222
15.3942
3.8930

Displacement (in)
16
Fig. 6-6 Center Displacement of Simply Supported Square Plate
under Distributed Load
138

Table 6-4 Comparison of Displacements of Square
Plate with Simple Support
Load
Load
Exact
ANSR
ERROR
(psi)
steps
(in)
(in)
(%)
0.000
0
0.0000
0.0000
0.0000
2.748
10
1.005
1.019
1.39
10.980
10
2.454
2.345
4.44
43.950
10
4.41
4.212
4.49
175.800
10
7.2
6.258
13.08
100
10.99
4.35
703.200
200
11.49
11.24
2.17
400
11.31
1.57
2813.100
200
18.21
17.80
2.25
The exact values are quoted from reference [20]

Table 6-5 Comparison of Displacements of Square
Plate with Clamped Support
Load
Load
Exact
ANSR
ERROR
(psi)
Steps
(in)
(in)
(%)
0.00
0
0.0000
0.0000
0.0000
5.337
10
0.711
0.7082
0.39
11.490
10
1.413
1.3960
1.20
20
1.4020
0.78
19.020
10
2.085
2.041
2.11
20
2.057
1.34
28.500
10
2.736
2.649
3.18
20
2.679
2.08
40.470
20
3.363
3.274
2.65
55.200
20
3.969
3.841
3.22
73.500
20
4.563
4.396
3.66
95.400
20
5.142
4.931
4.10
120.600
20
5.706
5.436
4.73
The exact values are quoted from reference [20]

Table 6-6 Displacements of Square Plate with
Fixed Support under Distributed Loads
Using Different Nonlinear Stiffnesses
without Iterations
step
no.
wt
(pcf)
GEO
(psi)
LD
(in)
ALL
(in)
0
0
0.0000
0.0000
0.0000
1
1000
0.2524
0.2524
0.2524
2
2000
0.5023
0.5030
0.5005
3
3000
0.7474
0.7483
0.7396
4
4000
0.9859
0.9854
0.9664
5
5000
1.2160
1.2120
1.1790
6
6000
1.4830
1.4280
1.3780
7
7000
1.6500
1.6320
1.5640
8
8000
1.8540
1.8250
1.7360
9
9000
2.0480
2.0060
1.8970
10
10000
2.2330
2.1780
2.0480
11
11000
2.4110
2.3400
2.1900
12
12000
2.5810
2.4930
2.3240
13
13000
2.7430
2.6390
2.4500
14
14000
2.8990
2.7770
2.5690
15
15000
3.0490
2.9080
2.6830
16
16000
3.1930
3.0330
2.7910
17
17000
3.3320
3.1530
2.8940
18
18000
3.4660
3.2680
2.9930
19
19000
3.5950
3.3780
3.0880
20
20000
3.7200
3.4840
3.1790

Displacement (in)
Fig. 6-7 Effects of Different Nonlinear Stiffnesses
on Clamped Square Plate under Distributed Load
142

CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
Two finite elements have been developed and implemented
in ANSR [56] for the analysis of a hollow box girder for
post-tensioned construction.
The three-dimensional link element used to model a dry
joint has shown realistic element behavior. It opens under
tension and closes under compression. The link element has
exhibited a cyclic convergence behavior.
The linear Mindlin shell element used to model the
concrete section of the hollow girder has shown an excellent
response within its small displacement assumption.
The nonlinear Mindlin shell element has been developed
to predict large displacement and initial stress (geometric)
nonlinearities. It has been derived directly from the
linearized incremental equilibrium equation. This is
basically a linear equilibrium equation within each
increment. Thus the formulation is similar to that of linear
element. The total Lagrangian formulation was used for the
description of motion. The disadvantage of this formulation
is that it needs special treatment for large rotations
because finite rotation is not a tensorial (vector)
quantity. One of the solutions to this limitation is co-
143

144
rotational formulation which is basically an updated
Lagrangian formulation for rotations only [26, 27]. The
displacement dependent loads, which are usual for shell
structures, is also recommended for further study.
Another area of further research is the material
nonlinearity of concrete. This nonlinearity is essential to
provide realistic structural response for safe and cost
effective designs. Some of the complex concrete properties
are nonlinear material properties, cracking in concrete,
shear transfer across cracked interfaces, time dependent
effects such as creep, shrinkage, and transient temperature
distribution [60].
The nonlinear material properties of concrete have long
been the subject of research. The first reliable test
results on the two dimensional constitutive relationship
were reported by Kupfer and Hilsdorf [61, 62] and were used
to develop the equivalent uniaxial strain concept [63].
Various constitutive models of concrete can be found in [60,
64].
An entirely different approach, endochronic theory, to
materials in which the inelastic strain accumulates
gradually was also suggested. It consists of characterizing
the inelastic strain accumulation by a certain scalar
parameter, called intrinsic time, whose increment is a
function of strain increment [65]. The cracks developing in
concrete have been studied for a long time. Theories and

applications have been developed for the mechanical behavior
of individual crack [66, 67], discrete parallel continuous
crack [68], distributed (smeared) in a fixed directions
cracks [60, 69, 70, 71] , distributed rotating cracks[72].
The smeared crack theory and fracture localization in
concrete is well documented in [73]. In a cracked reinforced
concrete flexural member, the intact concrete between each
pair of adjacent tensile cracks assists the tensile steel in
carrying the internal tensile force, and therefore
contributes to the overall bending stiffness of the member.
This is called tension stiffening [74], The shear transfer
through aggregate interlock was described in [75, 76, 77,
78]. The combination of all the effects from cracks,
aggregate interlock, dowel action, and tension stiffening in
concrete response can be found in [79].
It is well known that there can be numerical
instability, and sensitivity on finite element mesh size, in
the solution process for the strain softening material [80].
This has been overcome through the shear band concept [80]
or a specific element formulation, for example, the four
node isoparametric element suitable for modelling cracks in
[81].
All these complex nonlinear material properties of
concrete must be incorporated for the realistic analysis of
any concrete structure including the hollow box girder for
bridge structures.

APPENDIX A
IMPLEMENTATION OF LINK ELEMENT
The three dimensional link element was implemented to
the ANSR-III program. ANSR requires that ten subroutines be
written for an element to be used by the program [56], The
following is a description of the subroutines written for
the link element; INEL06, STIF06, STAT06, RINT06, EVNT06,
OUTS06. The subroutines RDYN06, INIT06, MDSE06, EOUT06, and
CRLD06 are not used by the current version of the link
element. As a result, dummy subroutines that consist of
return statements are written for these subroutines.
The element group number is set to six for the link
element.
A.1 General Implementation Details
Implicit double precision is used throughout the
interface subroutines.
IMPLICIT DOUBLE PRECISION(A-H,0-Z)
The labeled common block /INFGR/ for element group
information is defined as follows.
COMMON /INFGR / NGR,NELS,MFST,IGRHED(10),NINFC,LSTAT,
LSTF,LSTC,NDOF,DKO,DKT,EPROP(3,200)
146

where,
NGR
= Element group number.
NELS
= Number of elements within the group.
MFST
= Element number of the first element in
this group.
IGRHED(IO) = Element group heading.
LSTAT
= Length of words of state information
variables in /INFEL/.
LSTF
= Length of words of stiffness information
variables.
LSTC
= Length of words of stiffness control
variables.
NINFC
= Length of words of variables in /INFEL/.
= LSTAT + LSTF + LSTC
NDOF
= Number of element degrees of freedom.
DKO
= Initial stiffness damping factor.
DKT
= Current tangent stiffness damping factor/
DKO, DKT are not used in the link element but are
included in /INFGR/ because these are used by the base
program.
The labeled common block /INFEL/ for element
information is defined as
COMMON /INFEL / IMEM,KST,LM(12),NODE(5),MAT,CDEL,XL,
XX(5),YY(5),ZZ(5),T(3,3),XKS,XKN,XKSO,
XKNO,XNU,THK,DISU,DIST,DISV,DISW,STRU,
STRV,STRT,STRM,IJTS,UTO,FKP(78)
where, state information variables are IMEM through STRM.
The length of the state information variables in terms of
integer words, LSTAT, is 100 for this case because there are

twenty integer variables (IMEM through MAT) and forty real
variables (CDEL through STEM). A real word has a length of
two integer words. IMEM, KST, and LM(12) must be at the
beginning of the labeled common block /INFEL/ in the given
sequence. LM is 12 long since there are 12 element degrees
of freedom for the link element.
IMEM
=
Element number.
KST
=
Stiffness update code.
LM(12)
=
Location matrix.
NODE(5)
=
Node numbers, no. of 4-el nodes & K node.
MAT
=
Material property number.
CDEL
=
Allowable compressive displacement for the
passing of two joint nodes.
XL
=
Element length.
XKS
=
Current element shear stiffness.
XKSO
=
Original element shear stiffness.
XKN
=
Current element normal stiffness.
XKNO
=
Original element normal stiffness.
XNU
=
Friction coefficient.
THK
=
Element thickness.
DISU
=
Element deformation in u(x) direciton.
DIST
=
Element deformation in t(y) direction.
DISV
=
Element deformation in v(z) direction.
DISW
=
Element rotation.
STRU
=
Element stress in u(x) direction.
STRT
Element stress in t(y) direction.

149
STRV
= Element stress in v(z) direction.
STRM
= Element moment which shows the distribution
of element stress.
XX (5)
= X coordinates of four nodes and the K node.
YY(5)
= Y coordinates of four nodes and the K node.
ZZ (5)
= Z coordinates of four nodes and the K node.
T(3,3)
= Transformation matrix. (3,3) submatrix.
The stiffness control information variables are
UTO,IJTS. The word length of these variables, LSTC, is two.
IJTS
= INDEX FOR JOINT SLIDING.
(1 = SLID, 0 = NOT SLID)
IJTO
= INDEX FOR JOINT OPENING.
(1 = OPENED, 0 = CLOSED)
The stiffness information variables are FK(78). The
word length of this group, LSTF, is 156. The actual size of
the element stiffness matrix is twelve by twelve which
contains 144 real numbers. Because of the symmetry of the
element stiffness, only the lower half of the element
stiffness matrix, including the diagonal is to be saved. The
number of terms in that portion of the element stiffness
matrix is 78.
FK(78) = Element stiffness matrix.

A.2 Subroutine INEL06
Subroutine INEL06 is the input subroutine. Its purpose
is to read and print the input data for link elements and to
initialize the variables in the labeled common blocks
/INFGR/ and /INFEL/. This subroutine has the form of
SUBROUTINE INEL06 (NJT,NDKOD,X,Y,Z,KEXEC)
In the inel06 subroutine, the element group control
variables are to be set. The first one is MFST, the element
number of the first element in this group. MFST is input
data read by the base program. It is defaulted to be one at
the beginning of the inel06 subroutine if it is not given in
input file.
The index for the stiffness change, KST, is set to one.
This indicates to form the stiffness since this is the first
pass. KST can be updated in accordance with the actual
status of element stiffness, i.e., if element stiffness is
not changed KST is assigned the value of zero.
The length of the element information and the number of
element degrees of freedom are also to be set in inel06
subroutine.
LSTAT = 100
LSTF =156
LSTC = 2
NINFC = 258
NDOF
12

Then element group information is printed. NGR, IGRHED,
NELS, MFST are read in base program. All others are set in
inel06 subroutine.
The number of material properties and the material
properties are read and then echo-printed. CDEL is the
compressive deformation limit to define joint overlap. This
is the amount of displacement overlap allowed before the
stiffness is increased to prevent the loss in forces during
transfer through displacement. NMAT is the number of
material properties. For the link element, three material
properties are required, i.e., joint normal stiffness, joint
shear stiffness, and coefficient of friction.
An index for the error, INERR, is set up to detect an
error during the data input in inel06 subroutine. It is
initially set to zero. If an error is detected it is set to
one.
The control information for element data is calculated.
IMEM is the element number, which will be increased by one
after the end of each element data line is read or
generated. NLAST, the the last element number in this group,
calculated by taking the first element number plus the
number of elements. An index to check the number of the
lines of element data, ICNTR is set up. Element data are
read element by element. The required element data are
NEL
= Element number.

NODE(5)
MAT
= Numbers of four nodes and the third node.
= Material property of the element.
THK = Thickness of the element.
NGEN = Number of elements to be generated
including the element specified. (NGEN-1)
elements are to be generated.
KINCR = Increment of the number of the third node
for element generation.
If no generation of element data is specified, i.e.,
NGEN is zero. The numbers of node three and four are
interchanged for ease of input and then element data will be
processed by the subroutine EVEL06.
The node numbering for the formulation and data input
is
4 * * 3
1 * * 2
Node numbering for
formulation.
3 * * 4
1 * * 2
Node numbering for
data input.
If generation of element data are requested, i.e., positive
NGEN, (NGEN-1) sets of element data will be generated
followed by the element data processing. Before element data
are processed by the base program, all element data is
printed. Element data processing is performed by calling the
subroutine EVEL06 which is part of the element input
subroutine.
All these processes are done for each link element. For
the last element, the element number is checked against the

153
last element number NLAST. If there is a discrepancy, KEXEC
is set to one for data check mode and the subroutine EXIT is
called to terminate the program.
Element data processing is composed of
1. Continuous element numbering from the first MFST.
2. Fill LM array.
3. Form transformation matrix.
4. Initializing variables in /INFEL /.
5. Compute stiffness matrix profile.
6. Transfer element data to tape.
Continuous element numbering is done simply by
increasing the element number IMEM by one at the end of each
element processing.
The LM array contains the global degree of freedom
numbers corresponding to the element degree of freedom
numbers. The subroutine NCODLM must be called as many times
as the number of the terms in LM array. This subroutine is
called 12 times for the link element, since there are 12
element degrees of freedom.
ANSR numbers all the structural degrees of freedom in
the x-direction first and then those in the y-direction and
in the z-direction, respectively.
The transformation matrix is formed as follows:
1. Dimension a (3,3) matrix, i.e., T.
2. Find the element direction vector({i*}, local
u-direction vector) from the coordinates of the
start and end nodes and then normalize it by

154
calling VECTOR. Put the three components into the
first row of the [T] matrix.
3. Form the third direction vector({kk), local third
direction vector) from input by calling VECTOR.
This is the local n direction vector for the link
element. Therefore, normalize it({k'}) and put the
three components into the second row of the [T]
matrix.
4. Perform the vector cross {i* > x {k•> to find local
w-direction vector and normalize it({j *)) by calling
CROSS. Put the three components into the third row
of [T] matrix.
5. The resulting matrix is the transformation matrix
[T]. The length of the element is also calculated
during the formation of the transformation matrix.
The profile of the stiffness matrix is updated by
calling the subroutine BAND. The element data are
transferred to tape by calling the subroutine COMPAC.
A.3 Subroutine STIF06
Subroutine STIF06 is the element stiffness formation
subroutine. This subroutine is called whenever the
structural stiffness is to be formed or modified. If the
total element stiffness matrix is to be formed, the
subroutine is called once for each element. If a change in
stiffness is being formed, the subroutine is called only for
those elements which have undergone a stiffness change. The
total stiffness or change in stiffness must be returned in
array FK for assembly into the structure stiffness.

This subroutine has the form of
SUBROUTINE STIFO6(ISTEP,NDF,CDKO,CDKT,FK,INDFK,ISTFC)
The variables used in this subroutine are
FK = Element stiffness matrix(NDOF,NDOF).
FKG = Global element stiffness matrix.
FKL = Local element stiffness matrix.
FKP = Previous global element stiffness.
ISTFC = Stiffness matrix content index.
1 = Total element stiffness matrix.
0 = Change in element stiffness matrix.
INDFK = FK storage index.
1 = Lower half, compacted columnwise.
0 = Square(all).
ISTEP = Current step no. in step-by-step integration.
NDOF = Number of element degrees of freedom.
The index for the stiffness storage scheme, INDFK, is
set to zero in this routine since square storage compacted
column-wise is used.
The index for stiffness matrix content , ISTFC,
indicates whether the total stiffness matrix or the change
in stiffness is needed. This is sent in by the base program
and not set in the subroutine.
The linear part of element stiffness is formed and
rotated to global coordinate system.
There are two options for element stiffness formation.
If total element stiffness is requested, newly formed
element stiffness FKG will be transferred to FK. Otherwise,
the change in element stiffness is calculated by FKG minus
FKP and then this change will be transferred to FK.

The element stiffness in the local coordinate system is
formed in the subroutine LSTF06. The element stiffness
matrix is initialized with zeros and then the element
properties are updated. Each term of the upper triangular
and diagonal of the element stiffness is evaluated and then
lower triangular of the element stiffness is filled using
symmetry.
The transformation of the element stiffness from local
coordinates to global coordinates is done in the subroutine
TRAN06 using the three by three submatrix. This is done for
the efficiency and storage savings. The 12 by 12 element
stiffness matrix is divided into three by three submatrices
and the rotation is performed for each submatrix. This will
eliminate the unnecessary multiplication of zeros while
saving storage slightly.
The process is FKG = TT*FKL*T using 3x3 submatrices
1. Zero FKG(K,L).
2. Divide FKL(12x12) into 3x3 submatrix.
3. Zero TEMP3(3,3) for each manipulation.
4. TEMP3=TT*FKL
5. FKG=(TT*FKL)*T
A.4 Subroutine STAT06
Subroutine STAT06 is the state determination
subroutine. Its purpose is to update the element state

157
information in /INFEL/, given the current state and the
increment of nodal displacements in the global axes(array Q)
set up by the base program.
The subroutine has the form of
SUBROUTINE STAT06 (NDF,Q,TIME)
The variables used in this subroutine are
= INCREMENTAL global nodal displacements.
Therefore, if total stresses are needed,
the element displacements are to be added
up at the end of each iteration.
= INCREMENTAL local nodal displacements
= Incremental element deformation.
These are to be saved in infel for use
in rint06 to find FE for the equilibrium
check of incremental external loads.
= total elemental deformation,
where * = U,T,V, or W
If geometric (large displacement) nonlinear analysis is
requested, i.e., KGEOM is one, Coordinates of all the nodes
are to be updated and the new lengths of the elements with
new direction cosines are to be calculated and then the
transformation matrix is to be updated. Otherwise skip to
the small displacement analysis step.
For the analysis with the small displacement
assumption, the incremental global nodal displacements
(Q)(NDOF) are rotated to local coordinates (QDL)(NDOF). Then
the average incremental element displacements, DISUI, DISTI,
DISVI, DISWI are calculated. These incremental element
Q(NDF)
QDL(12)
DIS*I
DIS*

158
displacements are then added up to form total average
element displacements, DISU, DIST, DISV, DISW. The total
average element stresses, STRU, STRT, STRV, STRM are then
calculated from the total average element displacements.
These total stress are used for the state determination
along with the total average element displacements.
XNU is the static friction coefficient and XNUK the
kinetic friction coefficient. XKNU is assumed to be 0.75
times XNU. If the joint has slid previously (xks=0.),
kinetic friction stress has developed. This stress should be
incorporated in the internal resisting force recovery for
the equilibrium check.
Sliding is defined in three categories:
1. Sliding in u-direction.
2. Sliding in v-direction.
3. Sliding in both u- and v-direction.
If any one of these happens, the shear stiffness is set to
zero. The frictional force is treated as the unbalanced
force in the corresponding direction for the next iteration.
Joint sliding is defined as the state where the
vectorial sum of the element stresses in u- and v-directions
is greater than the magnitude of the frictional stress if
the normal stress is compressive. If the normal stiffness,
XKN, is zero, then no shear stress will develop, which was
considered in average stress calculation above. The normal

stress is normal stiffness, XKN, times the corresponding
displacement, DIST. The shear stiffness, XKS, is set to zero
whenever the normal stiffness is set to zero.
Before checking deformations, the stresses and
determining the state of the element, the stiffness change
index, KST is set to zero. If any change in element state
occurs, this index will be set to one so that the element
stiffness can be updated in the next iteration.
The change in element state is checked against four
possible cases. The element state change modes are
1. Closed to open.
2. Open to closed.
3. Stopped to sliding.
4. Sliding to stopped.
If none of these four changes in state occurs, there is no
change in element state and the index KST remains zero.
The state 'open' is defined is defined as
1. Avg. normal deformation dist> or = zero.
2. or strt > or = zero.
If a joint is opened, then there will be no shear and
no normal stiffness. This is indicated by setting the joint
opening index, IJTO, to one. The normal and shear stiffness
are also set to zero for the next iteration in the
subroutine STIF06. If these are to be set to zero here this
will affect the following decision statements in the rest of
the STAT06 routine. As a result, SKX is used as an argument

160
in the decision if-statement. XKS is the shear stiffness at
the beginning of the STAT06 routine.
'Overlap' is defined as the state where normal
compressive displacement is algebraically less than the
negative value of the given limit of compressive
displacement. This value, CDEL, is a positive number in the
input data. If joint nodes have overlapped beyond the limit
specified by CDEL, reactivate the shear stiffness XKS and
increase the normal stiffness XKN to prevent overlapping in
the next iteration. If (abs(dist).ge.cdel) is used, a large
tension disp will be taken as joint overlap, which is not
true.
If a joint has been closed without overlapping, the
normal stiffness and the shear stiffness will be reset to
the original values saved in the labeled common block
/INFEL/.
The element state 'slid' is defined as
sqrt(stru**2+strv**2) > -xnu*strt if strt < 0.
The case where the normal stress STRT is greater than
or equal to zero was covered in the joint opening decision.
The decision of whether the displacement,
sqrt(disu**2+disv**2), is greater than zero cannot be used
as a definition of sliding. This is because there is a
slight displacement before sliding occurs. This displacement
is not precisely known for every case to be analyzed.

If the joint is sliding, then no shear stiffness is
maintained. If the joint is sliding in one direction, it is
assumed that the joint is sliding in both directions. The
joint sliding index, IJTS, is set to one. The shear
stiffness, XKS may be set to zero because XKS is not used in
anymore if-statement. For the consistency in the program
structure, this is done in the subroutine STIF06, the same
as XKN.
If the joint is in the state of 'stop' which has
occurred from the 'sliding' state, shear stiffness is
recovered and IJTS is set to one.
The state determination for creep strain and large
displacement is to be done here. These are not included for
the current version of the link element.
A.5 Subroutine RINT06
Subroutine RINT06 is the element force recovery
subroutine. This subroutine is called for each element at
the beginning of the analysis and after each state
determination. Its purpose is to compute the element forces,
i.e., the nodal loads which are in equilibrium with the
current state of stress. These may be stated as equivalent
nodal loads which can cause the current state of element
stresses. Therefore, if these equivalent nodal loads are
subtracted from the actual nodal loads applied, current

unbalanced nodal loads will be obtained.
This subroutine has the form of
SUBROUTINE RINT06 (NDF,Q,VEL,FE,FD,TIME)
The meaning of the variables used in this subroutine
are
Q = Nodal displacements.
VEL = Nodal velocity.
FD = Dynamic nodal force when TIME > 0.
FE = Elasto-plastic nodal force when TIME = 0.
= Nodal loads which is in equilibrium with current
state of stresses.
= Imaginary forces which act on element to
introduce current element stresses.
The diagram for the element stress, element force,
internal resisting force and the external load is
<—°
> O
ELTMT
0
o m
*
<—0
0
*
EL
STRESS
FE
RI
NODE
element force internal external
resisting force
force

163
The sign convention for the element shear stresses is
A<
I I
I I
>v
positive shear stress
The recovered element forces which are in equilibrium
with the element stresses can be turned into internal
resisting forces by changing the direction of the forces.
These internal resisting forces are not usually in
equilibrium with external forces. The difference is the
unbalanced forces. In ANSR, the norm(sum of squares of each
difference) is used as decision variables for convergence.
For the equilibrium check, ANSR uses the total loads.
Therefore, the element forces are to be recovered using the
total element displacement. The element forces recovered
from the total element displacements are in the local
coordinates. These must be rotated to global coordinates so
that these forces can be compared with the global loads for
the equilibrium check. The rotation from the local to global
is done by
(local) = [T](global)
(global) = [T transpose](local)
The recovered element forces are transferred to the base
program through the FE array after the rotation from local
to global coordinates.

The nodal forces due to damping are supposed to be
calculated in this subroutine. The current version does not
include damping. If damping is considered, the variable TIME
will be greater than zero. This is compared with the
constant zero to see if damping is included.
A.6 Subroutine EVNT06
Subroutine EVNT** is called for each element at
frequent intervals during the analysis. Its purpose is to
calculate the proportion of the displacement increment, Q,
which can be applied to an element before a significant
nonlinear event occurs. Typical events are yielding, gap
closure, and unloading, i.e., the intersection of two linear
portions in structural stiffness.
This subroutine has the form of
SUBROUTINE EVNT06 (NDF,Q,VEL,ACC,EVFAC,IEV)
The variables used in this subroutine are
IEV = Event type index.
0 = No event.
1 = Link opened.
2 = Link slid.
3 = Link closed.
EVFAC = Event factor
= Force used to cause the event / Force applied.
If no event happens, EVFAC = 1.
The large displacements nonlinear analysis is not
included for the current version of the link element.

165
The number 0.999999 is defined as PTNINE to avoid the
numerical difficulty in comparing the calculated ratio to
one to predict an event.
Small displacements are assumed for the current version
of the link element. The nodal displacements are more useful
than the average displacements at the center of the element
because nodal displacements can be used to define the joint
closing to avoid passing of the nodes by providing higher
stiffness at the point of joint closing. The element nodal
displacements are recovered from the given global element
nodal displacements through rotation from global to local
coordinates.
If several events happened, the smallest event factor
governs. The events are identified in sequence and the
corresponding event type assigned. Then the event factor is
calculated and compared with the current smallest event
factor.
At the beginning of the decision process, the event
type is set to zero and the event factor to one. Two
additional variables are required as input data to define
events. These are the element normal and shear forces at the
opening event and sliding event, respectively.
The first event is a link opening event. This is
defined by the normal element displacement and the normal
stiffness. The previous contact state is identified by the
nonzero normal stiffness. The opening is identified by the

166
ratio of the normal displacement at the event over actual
normal displacement.
The second event is link sliding. The absolute shear
displacement, DIS, is calculated by the square root of the
sum of the squares of the displacements in local x and y
direction. The absolute displacement is then compared with
the shear displacement at the event.
The third event is link closing. If this happens, the
event factor is set to a very small number because there is
a sudden change in element behavior. The analysis process
needs to set back to the closing point and the normal
stiffness set to its input value.
A fourth event is node overlap. In this event the event
factor is given a very small number and the normal stiffness
is set to a higher value by a trial factor of ten to avoid
overlap in the next iteration.
A.7 Subroutine QUTS06
Subroutine OUTS06 is the output subroutine. Its purpose
is to print the current element stresses, strains, and
status information from the information in /INFEL/. The
subroutine has the form
SUBROUTINE OUTS06 (KPR,TIME)

If no print is requested, KPR is set to zero, return to
the base program. For the first element, KHED, is set to
zero to write heading, print index, and print request type.
For all subsequent elements, only the element information is
written. This is identified by the KHED. If KHED is not
zero, this element is not the first element.
The current element information printed in this
subroutine is element number, element node numbers, and the
element stresses in local u, n, and w directions.

A.8 Link Element Data Input Guide
(1) Control information
(1.1) First
control line
COLUMNS NOTE NAME
DATA
4- 5(1)
NGR
Element group number = 6
6-10(1)
NELS
Number of link elements
11-15(1)
MFST
Element number of first link
element
16-25(F)
DKO
Initial stiffness damping factor
26-35(F)
DKT
Tangent stiffness damping factor
41-80(A)
Optional heading for link element
(1.2) Second
control
line
COLUMNS NOTE NAME
DATA
1 -5(1)
NMAT
Number of material properties
6-15(F)
CDEL
Allowable compressible deformation
(2) Material
property
data
NMAT lines
COLUMNS NOTE NAME
DATA
1- 5(1)
MAT
Material property number
6-15(E)
XKS
Element shear stiffness
16-25(E)
XKN
Element normal stiffness
26-35(F)
XNU
Friction coefficient
(3) Element
data
NELS lines
COLUMNS NOTE NAME
DATA
1- 5(1)
NEL
Element number
6-10(1)
NODE1
Node 1
11-15(1)
NODE 2
Node 2
16-20(1)
NODE 3
Node 3
21-25(1)
NODE4
Node 4
26-30(1)
NODEK
Node K for transformation
31-35(1)
MAT
Material property number
36-45(F)
THK
Element thickness

46-50(1)
51-55(1)
NGENX
KINCR
Number of elements generated
Node K increment for el. generation
NOTES
(1) Local coordinate system
N0DE1,NODE2,NODE3,NODE4,NODEK
n
K
o A
I I
3 I | 4
o—
- - 1 —
o
O""'
/
/
2
w
Note: Local w axis is decided by right-hand rule coming
out of the page.

APPENDIX B
IMPLEMENTATION OF LINEAR SHELL ELEMENT
The implementation is based on the linear version of
three to nine node shell element from the simple analysis
(SIMPAL) program written by Dr. Marc I. Hoit. Isotropic
materials are assumed and arbitrary orientation of element
in global coordinate system is considered.
The shell element was implemented in the ANSR-III
program. The subroutines written are; INEL13, EVEL13,
VECT13, TLIST1, TLIST2, SELFWT, FRMST1, FRMST2, STIF13,
SHSTF1, SHSTF2, ELAW1, ELAW2, FORMH, GD, TRANS, TRIPL,
STAT13, RINT13, 0UTS13. The dummy subroutines for current
shell element are; RDYN13, CRLD13, E0UT13, EVNT13, MDSE13.
These are required by ANSR-III but not used for shell
element. Thus, all the dummy subroutines have only return
statements.
Implicit double precision was used for all the
subroutines.
B.l Subroutine INEL13(NJT.NDKOD.X.Y.Z.KEXECn
This subroutine reads element data for two dimensional
shell element. The element is formulated through the
170

combination of membrane element and plate bending element.
The assigned element group number is 13.
Named common blocks and the variables are as follows.
COMMON/INFGR/NGR,NELS,MFST,IGRHED(10),NINFC,LSTAT,LSTF,LSTC
NDOF,DKO,DKT,PROP(4,225),RS(2,8),RS4(2,4),WG(8),WG4(4),NMAT
ITMPLD,ALPHA,REFTEM,ITMOFF,IDUM(121)
NGR = Group number for current elements = 13 for shell
NELS = Number of current elements.
MFST = Element number of the first element in current
group.
IGRHED(10) = Element group heading.
NINFC = Length of common block /INFGR / in terms of
integer words.
LSTAT = Length of state information.
LSTF = Length of stiffness control information.
LSTC = Length of stiffness information.
NDOF = Number of element degrees of freedom.
DKO = Initial stiffness damping factor.
DKT = Current tangent stiffness damping factor.
DKO, DKT are not used for shell element but are
included in /INFGR / for the compatibility with base
program.
The element property array EPROP(IOOO) in the labelled
common block /infgr/ of base program has been divided into
small blocks of group information.
PROP(4,225) = Element property array which carries
Young's modulus, Poisson's ratio,
shear modulus and self weight. There is
a limit of 225 on the total number of
element property sets.

RS (2,8) and RS(2,4) = The coordinates of Gauss points
in local rs-systexn for the eight point
and four point numerical integration,
respectively.
The numerical integration is needed in the evaluation
of element stiffness, initial loads, and stresses. Selective
numerical integration has been used.
WG(8) and WG4(4) = Numerical Integration weights.
NMAT = Total number of material properties.
ITMPLD = Temperature load index for input.
1 = Input temperature of top and bottom
sides. The same temperature over the top
or bottom sides.
2 = Input different temperatures at each node.
ALPHA = Coefficient of thermal expansion.
REFTEM = Reference temperature for temperature change.
This is needed for the calculation of in-plane
thermal strains.
ITMOFF = Index to ignore the effect of temperature in
local r- or s- direction to model beam-type
structures.
1 = Ignore temp loading in r-direction(local x).
2 = Ignore temp loading in s-direction(local y).
IDUM(121) = Dummy integer array to make the length of
the total element group property exactly 1000.
COMMON/INFEL/IMEM,KST,LM(54),N0DE(9),NN,MAT,NRIP,THK,XX(9),Y
Y(9),ZZ(9),TT(3,3),XY(2,9),XM(54),DISPT(54),SIG(24),TEMPLD(5
4),SELFLD(54),SIGI(24),ST(24,54),FKP(1485)
This named common block carries the following element
information. State information is IMEM through ST(24,54).
The length is 3298 integer words consisting of 68 integer
words and 1615 real words. There is no stiffness control
information for shell element. Stiffness information is

173
FKP(1485) with the length of 2970 integer words from 1485
real words.
IMEM = Current element number.
KST = Control variable for stiffness change.
1 = Stiffness has been changed.
0 = Stiffness has not been changed.
LM(54) = Location matrix that contains global degree of
freedom numbers corresponding to element
degree of freedom numbers.
NODE(9) =
Node numbers rearranged for formulation.
NN
Number of nodes (4 to 9) of the element.
NRIP
Number of numerical integration points.
MAT
Material property number.
THK
Element thickness.
XX (9)
Nodal coordinates in global x.
YY (9)
Nodal coordinates in global y.
ZZ (9)
Nodal coordinates in global z.
TT(3,3) =
Transpose of transformation matrix for the
local X-, y-, z- vectors.
XY(2,9) =
Local dimensions dx, dy to be used for the
calculation of Jacobian matrix.
XM (54) =
Nodal mass matrix (to be used for dynamics).
DISPT(54)
= Total nodal displacements.
SIG(24) =
Six stress components at four integration
points to be extrapolated to nodal stresses.
TEMPLD(54) = Equivalent nodal loads for temperature.
SELFLD(54) = Equivalent nodal loads for self weight.
SIGI(24) = Initial stresses to be subtracted from total
stresses to yield actual stresses.

174
ST(24,54) = Element stress recovery array for linear
version. Stresses can be recovered from
local element nodal displacements as
follows.
[stress] = [E][strain]
= [E][d][u]
= [E][d][f][q]
= [E][ B ][q]
= [ST ][q]
where,
[E] = Stress-strain matrix = [C] in the program.
[d] = Differential operator relating strain and [u].
[u] = Generic displacement at a point within an
element.
[f] = Displacement function.
[q] = Nodal displacement in local coordinate system.
= Displacement [u] at element nodes.
FKP(1485) = Upper triangular portion of element
stiffness or change in element
stiffness(54,54).
COMMON /WORK / NNODE(9),IDUM,IJG(2),TEM(9)
NNODE(9) = Element node numbers for read-in.
IDUM = Integer dummy variable to make the length even.
IJG(2) = Number of elements to be generated in local i-
and j- direction including the one specified.
TEM(9) = Temporary array for temperature information.
DIMENSION NDKOD(NJT,6),X(NJT),Y(NJT),Z(NJT),I0RD(9),
NODGEN(9),TNODE(2,9)
NDKOD = Index array of global degree of freedom numbers
corresponding to element degrees of freedom
numbers.
NJT = Number of joints(nodes) in structure.
X(NJT) = X-coordinates of all nodes.
Y(NJT) = Y-coordinates of all nodes.
Z(NJT) = Z-coordinates of all nodes.
IORD(9)= Order of node numbers for formulation.
NODGEN(9) = Node numbers for generation.

TNODE(2,9) = Top and bottom temperatures at nodes.
The array iord(9) has the order of element node numbers
used for the formulation which is different from the order
used for data read-in. This array will be used for the
rearrangement of element node numbers for formulation at the
time of data input. This array must be dimensioned because
the array is filled by data statement and thus cannot be
included in the common block.
Other variables used in this subroutine are
NODE(9) = ordered node numbers for formulation
WG = integration weights
IEL = number of elements to be generated in i-direction
JEL = number of elements to be generated in j-direction
INC = node number increment in i-direction for element
generation
JNC = node number increment in j-direction for element
generation
IJG(2) = number of elements to be generated including
the element specified in i (IJG(l)) and
j (IJG(2)) direction. Element generation works
only for rectangular elements.
The element number of the first shell element MFST is
set to one unless otherwise specified. The element stiffness
is currently constant for the linear shell element and
therefore the stiffness change index KST is set to zero.
The word lengths of element information groups are
figured out. Length of element state information variables
LSTAT, length of stiffness control variables LSTC, length of

176
stiffness information variables LSTF and total length of
common block /infel /, NINFC are calculated.
The number of element degrees of freedom in 3-D global
coordinate system NDOF is set to 54. There are six dofs per
node, i.e., three translations and rotations. Thus the total
number of element dofs is 54 dofs for nine nodes.
Default integration weight is set to 0.999 if optional
integration weight is not provided through input. The shell
element uses 8 point Gauss Quadrature. Integration constants
are calculated based on optional or default integration
weight and saved in WG(8) for numerical integration. AW is
the primary integration point calculated and is to be used
in the stress calculation.
The array of integration point coordinates in the local
system is built and saved in RS(2,8).
Four point integration parameters are calculated in
local coordinates and saved in RS4(2,4) with the
corresponding weights WG(4) for four point integration on
the plate element.
Element property set is read and written. This includes
material property number, Young's modulus, Poisson's ratio,
shear modulus and self weight. If shear modulus is not
given, assume isotropic material, so that
G = (1/2)*( E / (1 + 2*P0I))

The self weight was assumed to be given in lb/cf and
this is changed to kips/cubic inch because the shell element
is used basically to model the concrete box girder itself.
The element number will be added up at the end of each
element data line including the one to be generated. Thus,
let it be one less than the element number of the first
shell element.
The element number of the last shell element is
calculated. This will be compared with the element number of
the last element data to check if all the element data lines
have been read.
Element information is then read line by line. Each
element input line has an element number, material property
number, nine node numbers, thickness, number of elements to
be generated. The element thickness is defaulted to one if
not specified.
Temperature information is read using the temperature
input control variable ITMPLD. Temperature can be given in
two ways. If temperatures at top and bottom faces are
constant, then ITMPLD is one and temperature at top face and
temperature at bottom face is given in one line. If
temperatures at top and bottom faces are varying, ITMPLD is
two and the temperatures at each node for top and bottom
faces are given in two lines. The order of nodal
temperatures is then re-organized to match the one used for
formulation.

If element generation is specified, new elements are
generated as necessary. IJG(l) elements are generated in i-
direction, ijg(2) elements are generated in j-direction
including the element specified. Thus (IJG(1 or 2) - 1)
elements will actually be generated.
The order of element nodes for data input and
formulation is as follows:
7 8 9 4 7 3
4 5 6 -> 8 9 6
12 3 15 2
Node Numbering for
Data Input
Node Numbering for
Formulation
Once element data are read then they are processed for
each element by calling the EVEL13 subroutine.
If there remain more element data lines after the
processing of the previous element data, the next input line
is read. Otherwise, the last element number is checked for
termination of data input. In case of any error in element
data input, an error message will be written and then the
program will be stopped.
B.2 Subroutine EVEL13(NJT.NDKOD.X.Y.Z.NODGEN.TNODE.
ICNTR,INERR)
Newly introduced variables are:
AA(4) = Direction cosines of r-axis( local x).

179
BB(4) = Direction cosines of s-axis( local y).
CC(4) = Direction cosines of t-axis( local z).
This subroutine processes element data element by
element. Element number is updated and the number of
elements processed is counted for input control. Dispt(54),
fkp(1485), selfld(54), lm(54) arrays are initialized.
The coordinates of four corner nodes are recovered from
the global coordinate arrays. Four corner nodes one, two,
three and four used for formulation and local element
coordinate system are shown in Fig. B-l.
The node number three is the origin for the local
coordinate system for the calculation of local i- and j- and
k-direction vectors. Local x-axis goes from node three to
node four. Local y-axis goes from node three to two. The
element i-vector, a unit direction vector in local x-
direction and the element j-vector, a unit vector in local
y-direction are formed directly from the coordinates of the
nodes by calling VECT13. The element k-vector, a unit vector
in local z-direction can be found by a vector cross.
k = i*j for right-handed coordinate system.
Local j vector must then be modified using the second vector
cross.
j = k*i
The transpose of transformation matrix can then be
assembled from the unit local coordinate vectors as follows.

180
3
dy(l)
Fig. B-l Local Coordinate System and Dimensions

181
i' = aa(l)i +
j ' = bb(l) i +
k' = cc(l) i +
i' aa(l)
j' bb(l)
k' cc(1)
aa(2)j
+
aa(3)k
bb(2)j
+
bb(3)k
cc(2)j
+
cc(3)k
aa(2)
aa(3)
bb(2)
bb (3)
cc(2)
cc(3)
i
j
k
[ local ] = [ T ] [ global ]
Local dimensions dx, dy for the current element are
calculated using the coordinates of start and ending points
and local coordinate vectors through vector dot product to
get the projection of the element dimension onto the element
coordinate system. This is shown in Fig. B-l. The number of
nodes in the current element is counted to collapse local
dimensions. The array XY(2,9) has been dimensioned for a
nine-node element. If any node is missing, the above process
is skipped. Thus the values of the next node are shifted to
the space for the current node, which is missing. These
dimensions will be used in the calculation of the elements
of Jacobian matrix. Refer to the subroutine FORMH.
Element location matrix is set up and the number of
integration points set. One point integration is used for 4
or less nodes and four point integration is used if the
number of nodes is greater than four.
The stress recovery array ST is formed for linear
stress recovery. [Stresses] equals [ST][q], where [q] is

182
element nodal displacements in local coordinate system. The
ST array relates the local nodal displacements to the
element stresses at the integration points. After
initialization of the ST array, it is formed by calling the
subroutines FRMST1 and FRMST2 for membrane and plate
portion, respectively.
Equivalent nodal loads for temperature loads and
initial stresses due to the temperature loads are
calculated. These initial stresses will be subtracted from
the stresses calculated using the total element
displacements because the temperature strain does not
introduce stresses. Sometimes the total strain is divided
into two components, i.e., mechanical strain and thermal
strain. Only the former produces stresses. The equivalent
loads must be subtracted from the internal resisting forces
formed by the total displacements or total stresses for
equilibrium check in RINT13 subroutine. These calculations
are done in the subroutines TLIST1 and TLIST2 for membrane
and plate elements, respectively.
This calculation is skipped if there is no temperature
loading using the temperature load index ITMPLD.
Equivalent nodal loads for self weight are calculated
by calling the subroutine SELFWT. These equivalent nodal
loadings are not temperature-type loadings even though these
are treated similarly in this subroutine. Thus the
equivalent nodal loads due to self weight will not be

183
subtracted from the internal resisting forces recovered from
the total displacements.
Element stiffness matrix profile is computed by calling
the subroutine BAND.
The element data processing is over after the data are
transferred to tape through the subroutine COMPAC.
B.3 Subroutine VECT13fV.XI.YI.ZI,XJ.YJ.ZJ)
This subroutine finds the unit vector of a given vector
specified by the coordinates of starting point and ending
point. V(4) is dimensioned and Xi, Yi, Zi are the
coordinates of the starting point of a given vector and Xj,
Yj, Zj are the coordinates of the ending point of a given
vector. The magnitude of the vector can be found by the
square root of the sum of the squares of the coordinate
differences in three global coordinate directions. The
magnitude is stored in V(4). The three components of the
unit vector can be found simply by dividing the coordinate
difference in corresponding direction by the magnitude and
are stored in V(l), V(2) and V(3).
B.4 Subroutine TLIST1(TNODE)
This subroutine forms equivalent element nodal loading
due to temperature differential in local coordinate system

for a membrane element and then rotates them into global
coordinate system.
COMMON /WORK / H(3,9),EB(3),B(3,18),PHI01(3),tem(54)
+
PHIOl(3) = Initial strain due to temperature loading
for membrane portion of shell element.
TEM(54) = Temporary array for the rotation of
temperature load to global coordinate system.
DIMENSION ini(18), in2(12), in3(18),tnode(2,9),cm(3,3)
DATA
in2 71,2,4,7,8,10,13,14,16,19,20,22/
in3 /l,2,7,8,13,14,19,20,25,26,31,32,37,38,43,44,49,50/
CM(3,3) = Constitutive matrix for membrane element.
The data in2 contain the numbers corresponding to in¬
plane stresses out of six stresses for four integration
points. This will be used in locating the stresses from
membrane portion in the 24 stresses possible for shell
element in global coordinate system.
The data in3 contain the numbers of in-plane membrane
element dofs out of 54 global dofs of shell element in
global coordinate system. This will be used the
transformation of membrane portion of equivalent nodal loads
into global loading which has a size of 54.
The number of columns of local membrane element strain-
nodal displacement matrix is set to the number of nodes
times two. For nine node element, this will be 18.
The arrays b(3,18), templd(54), tem(54) are
initialized. The element properties needed in this
subroutine are recovered. These are Young's modulus and

185
Poisson's ratio. The isotropic stress-strain law for
membrane elements is evaluate by a call to ELAW1.
The equivalent nodal loadings due to temperature
effects are calculated through the loop over all integration
points using numerical integration. The following procedures
are performed for each integration point and the results are
summed up.
The shape functions,their derivatives and Jacobian
matrix at current integration point are formed through
FORMH.
The weighting factor for the current integration point
including the determinant of Jacobian matrix and thickness
is calculated.
Then the strain-displacement matrix, b(3,18) is
calculated. Refer to frmstl for details.
The initial strain phiol(3) for the membrane portion of
the shell is computed using given temperatures. The membrane
has only the three in-plane strains out of the five strain
components of a shell element.
If uniform temperature differential is given, the
temperature differential to calculate initial strains for
the membrane element is the difference between average
temperature at neutral surface and the given reference
temperature.
The temperature difference for membrane strains at the
current integration point, DELTI, is sum of the difference

in each contribution of the temperature difference at each
node. This contribution can be found by multiplying the
numerical value of the shape function at the current
integration point and the corresponding temperature
difference at the node considered. Each difference is the
average temperature minus reference temperature for membrane
strains.
Otherwise, the difference is calculated and multiplied
by the corresponding shape function for the contribution of
the temperature differential for the current node. These
contributions are then summed up for all the nodes.
The in-plane initial strain array due to temperature
change PHI01(3) is initialized and calculated. The initial
strain can be calculated as the temperature difference times
the thermal expansion coefficient for the material
specified. The thermal strain component for in-plane shear,
Óxy, is zero because the temperature differential is the
same in local x- and y- directions. This means that no in¬
plane shear stresses will be introduced by the uniform
temperature differential.
Once the initial in-plane strains are calculated,
equivalent nodal loading and initial stresses can be found
through numerical integration.
Temperature loads are the negative values of those
recovered from the initial strain due to temperature loads
because equivalent load(54) is set to the negative values of

templd(54) in the subroutines ansr/static/templd.f and
ansr/load/elfrc.f.
Equivalent nodal loading is calculated through
numerical integration. The thickness term is included in the
integration weight. The numerical calculations for each
integration point are as follows.
The strain displacement matrix [B] is formed by
choosing proper terms from H(3,9) evaluated by a call to
FORMH subroutine. The equivalent nodal loads due to
temperature change are then obtained through the numerical
integration of [B]transpose*[E][phiol] over the volume of
the element. Here this integration is performed over the
area as the thickness is constant and has been included in
the integration weight. Local temperature loads are then
rotated to global coordinates for later assembly into global
loads.
The initial stresses due to thermal loadings can be
evaluated through the loop over the integration points. One
point or four point integration scheme is used depending on
the number of element nodes. Once the formation of membrane
element initial stresses in the local coordinate system has
been done, these will be subtracted from the element
stresses recovered from the total displacements in
subroutine STAT13. As the element stresses are calculated in
the local coordinate system, these initial stresses of
membrane element are placed at the corresponding locations

188
of the shell element initial stresses for possible
combination with plate element initial stresses. The
following processes are done for each integration point.
The shape functions, their derivatives, Jacobian, and
its inverse are numerically evaluated for the current
integration point. The initial strain PHI01(3) for the
membrane portion of the shell is formed. The initial stress
matrix for current Gauss point is calculated through the
numerical integration of [E][phiol] over the area.
B.5 Subroutine TLIST2(TNODE)
This subroutine forms the equivalent nodal loads and
initial stresses of plate element due to temperature loads
in local coordinate system and fills these into the shell
element temperature loads and initial stresses for the
combination with plane membrane temperature loads and
initial stresses.
COMMON /WORK / b(5,27),eb(5),h(3,9),phio2(5),tem(54)
PHI02(5) = Initial strain of plate bending element due
to temperature loading.
DIMENSION in2(20),in3(27),tnode(2,9),cp(5,5),c2(5,5)
CP(5,5) = Constitutive matrix of plate bending element.
C2(5,5) = Constitutive matrix with thickness terms
taken out for stress calculation.
DATA
in2 /l,2,4,5,6,7,8,10,11,12,13,14,16,17,18,19,20,22,23,
24/

189
in3 73,4,5,9,10,11,15,16,17,21,22,23,27,28,29,33,34,35,
39,40,41,45,46,47,51,52,53/
The data in2 contain the numbers corresponding to in¬
plane stresses and shear stresses out of six stresses for
four integration points. This will be used in locating the
stresses from plate portion in the 24 stresses possible for
shell element in global coordinate system.
The data in3 contain the numbers of plate element dofs
out of 54 global dofs of shell element in global coordinate
system. This will be used the transformation of plate
portion of equivalent nodal loads into global loading which
has a size of 54.
The number of columns of strain-nodal displacement
matrix for plate is set to the number of nodes times three.
In case of nine node element, this will be 27.
The isotropic constitutive matrix for plate element
CP(5,5) is formed by a call to subroutine ELAW2.
Element properties needed in this subroutine are
recovered. These are Young's modulus, Poisson's ratio and
shear modulus.
The equivalent nodal loads are formed through the loop
over integration points. The calculations for each
integration point are as follows.
The shape functions, their derivatives, Jacobian, and
its inverse are evaluated for the current integration point.
The strain-displacement matrix B(5,27) is then formed

through the choice of proper terms from H(3,9) matrix. The
initial thermal strain phio2(5) due to the temperature loads
is calculated. Temperature loads are the negative values of
those recovered from the initial strain due to temperature
loads because equivalent load(54) is set to the negative
values of templd(54) in the subroutines of base program
ansr/static/templd.f and ansr/load/elfrc.f.
Temperature difference at current integration point,
DELTI, is calculated in a similar manner with the membrane
portion of the shell element. But the difference between top
and bottom temperatures is used for the initial strains of
plate element. The initial strains for plate bending element
due to temperature loading can be found by temperature
difference times thermal expansion coefficient divided by
the element thickness. There are only two non-zero terms out
of five terms.
If temperature loading in x- or y- direction needs to
be ignored to model beam type structure, the corresponding
thermal strain is set to zero. This is for the comparison of
the results with those from beam theory for verification.
Once initial strains are calculated, equivalent nodal
loads due to thermal loading are evaluated through numerical
integration of [B]transpose[E][phio2] over the area.
The local temperature loads are then rotated to global
coordinates for later assembly into global loads.

The initial stresses due to thermal loadings can be
found through the loop over the integration points by
numerical integration as follows.
The initial stresses of the plate element due to
temperature loads in the local coordinate system are
computed, which will be subtracted from the recovered total
element stresses. These stresses are then placed at the
proper locations of the shell element initial stresses for
the combination with plane membrane stresses using data in2.
The initial stresses are calculated through the loop
over stress output points (integration points).
The calculation procedures are as follows:
General stresses are defined as moment resultant over
the thickness per unit length and thus have the unit of
moment per unit length while common stresses have the unit
of forces per unit area. The conversion can be done by
removing thickness terms from constitutive matrix. The
resulting matrix has been named C2(5,5).
The factor 1.5 is divided for maximum shear at center
line because the formulation gives only average transverse
shear stresses while the actual distribution is a parabola.
The remaining processes are identical with those for
membrane element.

B.6 Subroutine SELFWT
This subroutine forms equivalent nodal loads for the
self weight applied in the direction of gravity in global
coordinate system.
WGHT is self weight in local thickness direction per
unit thickness which was stored in prop(4,mat).
Self weight is divided into local rst-components using
the transformation matrix.
The direction of gravity in global coordinates is
defined as follows.
application direction X Y Z -X -Y -Z
IGRAVD 12 3-1 -2 -3
1 = T g
Let 1 be local components of self weight (br,bs,bt)
transpose, T be usual transformation matrix and g be global
self weight which has only one component in the direction of
gravity, for example, {0,by,0> transpose, where by is self
weight(wght). Therefore the relationship can be shown as
br = t(l,2)*wght (local x component)
bs = t(2,2)*wght (local y component)
bt = t(3,2)*wght (local z component)
As the transpose of transformation matrix has been
formed, the terms for transformation matrix are expressed by
the gravity direction index. The negative signs of local
gravity components are due to the assumption that the

gravity in global negative direction is defined as being
positive.
The equivalent nodal loads for self weight are
evaluated through the loop over integration points. The
procedures are as follows:
Determine the shape functions, their derivatives,
Jacobian, and its inverse numerically for the current
integration point and store them in h(3,9), the array of
shape functions h(l,i) = f¿ h(2,i) = f¿,x h(3,i) = (f^,y).
Integration weight is retrieved for the current
integration point. The equivalent local nodal loads due to
self weight are calculated by integrating local components
of gravity using shape functions and corresponding
integration weight. These are then transformed into a global
coordinate system using the transformation matrix.
B.7 Subroutine FRMST1
This subroutine forms the stress-nodal displacement
array [ST] for membrane element.
COMMON /WORK / H(3,9),B(3,18),CP(3,3)
H(3,9) are shape functions and their derivatives with
respect to global x and y for each node up to nine nodes.
node # 123456789
row #1 f^ ... f¿ ... fg
row #2 f1#x — fi,x — fg,x
row # 3 f1#y ... fify ... fg,y

B(3,18) is strain-nodal displacement matrix for
membrane element and cp(3,3) is constitutive law for
membrane element.
The data in2(12) give the locations of in-plane streses
from the membrane portion of the shell element out of 24
stress terms at four integration points. Each integration
point has six stress terms. There are three non-zero stress
terms from membrane portion of shell element for each
integration point. This shows the row number of [ST] array
for membrane stresses.
Stresses are calculated at each of the four integration
point. There exist three non-zero stresses for membrane
portion of shell element. For integration point number one,
the numbers for these non-zero stresses are
existing(membrane)
1
2
4
non-zero
stresses
XX
yy
xy
possible
stresses
XX
yy
ZZ
xy
yz
zx
1
2
3
4
5
6
The data in3(18) has the numbers corresponding to two
in-plane translational degrees of freedom at nine nodes out
of 54 degrees of freedom for nine node element. This shows
the column number in [ST] array for membrane stresses. The
numbers for node number one are
existing(membrane) 1 2
dofs x y
possible dofs x y z xx yy zz
in global system 123456

195
The number of columns in stress-nodal displacement
matrix is two times the number of element nodes for membrane
element. There are two translational degrees of freedom per
node.
The element properties, Young's modulus and Poisson's
ratio, are recovered to evaluate isotropic constitutive
matrix for membrane.
The stress-nodal displacement array [ST] can be
calculated through the loop over stress output points
(integration points).
[stress] = [E][strain ]
(6) = [E] [d] [u ]
(6.6)(6,3)(3,1)
= [E] [d] [f] [q]
(6.6)(6,3)(3,54)(54,1)
= [E][ B ][q]
(6,6)(6,54)(54,1)
= [st ][q]
(6,54) (54,1)
(numbers for one integration point)
For four integration points, the size is
[ST ][q ]
[24,54 ][54]
Six stress components at four integration points makes
a total of 24 stresses.
In general, the relationship between strain and generic
displacements can be shown as follows
[strain]=[differential operator][generic displacements]
e(ij)= (1/2)(u(i,j)+u(j,i)+u(k,i)u(k,j))

196
i.e.,
e(ll)=
e(22)=
e(33)=
e(12)=
e(23)=
e(13)=
/2) (u1#1+ulf
/2)(U2 2+u2,2+u1,22+U2,22+U:
./2) (Uo o+Uo ,+Ui 3^+Uo o -1-"
;+U,
2
i • ^2
2'U3'2;
)
)
)
+n3,lU3,2\
+U3,2U3,3¡
+U3,lu3,3)
This can be re-arranged as follows:
12 3
2e(ll)
2dll
0
0
2e(22)
0
2d22
0
2e(33)
0
0
2d33
2e(12)
=
d22
dll
0
2e(23)
0
d33
d22
2e(13)
d33
0
dll
ul
U2
U3
where d** is a differential operator.
Thus, the size of [d] is six by three in general.
The general relationship between generic displacements
and nodal displacements can be shown as follows:
12 3
ul
fll
u2
=
fll
u3
fll
qll
q22
q33
q44
q55
q66
(3,1) (3,54) (54,1)
where,
fll = {fl f2 f3 f4 f5 f6 f7 f8 f9}

197
and,
qll = {ql q7 ql3 ql9 q25 q31 q37 q43 q49} transpose
q22 = (q2 q8 ql4 q20 q26 q32 q38 q44 q50) transpose
q33 = {q3 q9 ql5 q21 q27 q33 q39 q45 q51) transpose
q44 = {q4 qlO ql6 q22 q28 q34 q40 q46 q52} transpose
q55 = {q5 qll ql7 q23 q29 q35 q41 q47 q53) transpose
q66 = {q6 ql2 ql8 q24 q30 q36 q42 q48 q54) transpose
This shows the size of [f] which is six by fifty-four.
The shape functions and their derivatives with respect
to to global x and y at the current integration point are
calculated by calling FORMH subroutine. For details, refer
to the subroutine FORMH.
The strain-nodal displacement matrix [B] is then formed
by selecting proper terms from H(3,9). In subroutine FORMH,
all the components of [B] matrix have been calculated. Here
the correct components from H(3,9) are simply placed at the
proper locations in [B].
b(3,18) = ^(3,2) , i=l,9
node # 1 ... i
row#l | fx,x 0 |
row#2 I 0 f1#y I
row#3 I flry flfx I
I 0 I
I o fify I
I fi»Y ffrX |
9
The stress-displacement matrix [ST] is equal to [E][B]
for membrane element and is placed at the correct places in
the shell [ST] using data in2 and in3.
The base index ii shows the location of row number for
current stress in data in2.

B.8 Subroutine FRMST2
This subroutine forms the stress-nodal displacement
matrix [ST] for the plate portion of the shell element.
COMMON /WORK / CP(5,5),C2(5,5),H(3,9),B(5,27),EB(5)
CP(5,5) is constitutive law for the plate bending
element for stiffness formulation and c2(5,5) is
constitutive law for plate bending element for stress
recovery. Thickness terms have been stripped. B(5,27) is
strain-nodal displacement matrix for the plate bending
element and eb(5) is an temporary array for [E][B]
calculation.
The data in2(12) give the locations of stresses from
the plate portion of shell element out of 24 stress terms at
four integration points. Each integration point has six
stress terms. There are five non-zero stress terms from
plate portion of shell element for each integration point.
This shows the row number of [ST] array for plate stresses.
Stresses are calculated at each of the four integration
points. There exist five non-zero stresses for the plate
portion of the shell element. For the integration point
number one, the numbers for these non-zero stresses are
existing(plate)
1
2
4
5
6
non-zero
stresses
XX
yy
xy
yz
zx
possible
stresses
XX
yy zz
xy
yz
zx
1
2 3
4
5
6

199
The data in3(18) has the numbers corresponding to one
out-of-plane translational degree of freedom and two out-of¬
plane rotational degrees of freedom of plate element at nine
nodes out of 54 degrees of freedom available for nine node
element. This shows the column number in [ST] array for
plate stresses.
The numbers for node number one is
existing(plate) 345
element dofs x y z xx yy zz
in global system 123456
The number of columns in stress-nodal displacement
matrix is three times number of element nodes for plate
element. There are three degrees of freedom per node.
After the arrays b(5,27), h(3,9),c2(5,5) have been
initialized, the element properties needed for constitutive
law are recovered. These are Young's modulus em, Poisson's
ratio poi and shear modulus G.
The isotropic stress-strain law for plate elements is
evaluated by a call to subroutine ELAW2. The thickness term
is removed from the constitutive matrix for stress
calculation of plate element.
The stress-nodal displacement array [ST] can be found
through the loop over stress output points in a similar way
with the procedures used for membrane portion of the shell
element by adding up of stress contribution for all the
integration points.

200
The procedures are as follows: the shape functions,
their derivatives, Jacobian, and its inverse are formed
first. The strain-displacement matrix b(5,27) is formed
using the proper terms from h(3,9).
b(5,27)
=
bi(5,3), i=
=1,9
node #
1
• • •
i
* • •
row#l
1
0 0
fl,x
1
• • • 1
0 0
fi,x
row# 2
1
o -£lfy
0
1
• • • 1
o -fj_,y
0
row# 3
1
o -f1#x
fi,y
1
* * * 1
o -fi,x
fi'Y
row# 4
1
f,,X 0
fi
1
* * * 1
fifx 0
fi
row# 5
1
fl'Y "fl
0
1
• t 1 1
fi'Y -fi
0
The stress-displacement matrix [ST] for current Gauss
point is calculated, added up and then placed into proper
places of stress recovery array [ST] of shell element using
data in2(row) and in3(column).
B.9 Subroutine STIF13(ISTEP.NDF,CDKO,CDKT.FK.INDFK.ISTFC^
This subroutine calculates the shell element stiffness
by combining the stiffnesses of both the membrane element
and plate bending element.
The shell element stiffness formed in the local
coordinate system is then rotated to the global coordinate
system. It is noted that stiffness is a second order tensor
and therefore it follows a different transformation law from
the one used for forces or displacements which are a first
order tensor (a vector).

[global stiffness] = [T transpose][local stiffness][T]
COMMON /WORK / S(45,45)
The dimension statement defines global shell element
stiffness FK, constitutive matrices of membrane and plate
elements, CM and CP.
DIMENSION FK(54,54),cm(3,3),cp(5,5)
FK = current global element stiffness matrix
FKL = S = current local element stiffness matrix
FKP = previous global element stiffness
FKC = current global element stiffness
ISTFC = stiffness matrix content index
(supplied by the base program)
1 = total element stiffness matrix
0 = change in element stiffness matrix
If the change in element stiffness is
local, the unchanged element stiffness can
be stored and only the changed portion
may be calculated and updated.
INDFK = FK storage index
1 = lower half, compacted column-wise
0 = square (all)
ISTEP = current step no. in step-by-step integration
>1 = form dynamic stiffness
1 = form dynamic stiffness at start of new phase
< 0 = form static stiffness
0 = form static (and geometric) stfns at start
NDOF = number of element degrees of freedom
The stiffness matrix storage scheme, INDFK is set to
zero. If the lower half compacted column-wise can be used,
it is better for the symmetric element stiffness matrix but
this causes some problem. Therefore INDFK is set to 0 for
square matrix storage scheme.

All the element properties are recovered for the
calculation of constitutive matrices. These are Young's
modulus, Poisson's ratio and shear modulus.
The calculation of MEMBRANE stiffness has been done as
follows.
The isotropic stress-strain law for the membrane
element is evaluated by calling the subroutine ELAW1. Then
the lower triangular portion of membrane element stiffness
is formed in the local coordinate system and placed at the
proper places of local shell element stiffness S(45,45).
This is done in the subroutine SHSTF1.
The calculation of plate contribution has been done in
a similar manner.
The isotropic stress-strain law for plate element is
calculated in the subroutine ELAW2. Then the lower
triangular portion of plate element stiffness in the local
coordinate system is formed and placed at the proper places
of local shell element stiffness S(45,45) in the subroutine
SHSTF2.
The upper part of local shell element stiffness matrix
is filled.
The local shell element stiffness is transformed from
local to global coordinate system by calling the subroutine
TRANS.
Transfer element stiffness information to base program
in accordance with the control variable ISTFC. This variable

is set to one for the first iteration for full element
stiffness. From the subsequent iteration, it is set to zero
only for the change in element stiffness by base program.
B.10 Subroutine SHSTFlfcn
This subroutine forms lower triangular portion of
element stiffness in local coordinate system for membrane
element and puts them into full local shell element
stiffness S(45,45). Five degrees of freedom per node times
nine nodes gives a total degrees of freedom of 45. And hence
the size of element stiffness matrix is 45 by 45.
COMMON /WORK / S(45,45)
C(3,3) = stress-strain law for membrane element
S(45,45) = full element stiffness matrix for shell
element.
Element stiffness is the integration of
([B]transpose[E][B]) over element volume.
Therefore the procedure will be as follows.
The shape functions, their derivatives and Jacobian
matrix are formed to evaluate strain-nodal displacement
matrix [B]. The element stiffness is calculated by numerical
integration of [B]transpose[E][B] over element volume using
the constitutive matrix brought in through C.
The data ini contain the numbers of degrees of freedom
of shell element which correspond to the degrees of freedom

of membrane element. These are two in-plane translational
degrees of freedom.
The loop over integration points for stiffness
calculation is as follows:
1) Form shape functions.
2) Calculate current integration weight factor.
3) Calculate strain-displacement matrix [B].
4) Perform Gauss quadrature on point r,s to form
stiffness matrix [B]transpose[E][B], where
[E]=[cm(3,3)]
5) Put them into the corresponding locations of full
shell element stiffness matrix using data /ini/.
B.11 Subroutine SHSTF2fC^
This forms the lower triangular portion of the plate
element stiffness in the local coordinate system and fills
these terms into the shell element stiffness.
The data ini is the numbers of degrees of freedom of
plate bending element in the shell element. These are one
out-of-plane translational degree of freedom and two out-of-
plane rotational degrees of freedom.
The procedures are the same as those of the membrane
element except for the size of the [B] and [E] matrices.

205
B.12 Subroutine ELAWlfEM.POI.CM
This subroutine evaluates isotropic stress-strain law
for membrane elements.
cm(3,3) = constitutive matrix
em = Young's modulus
poi = Poisson's ratio
Temperature effects on em and/or poi ignored. Initial
em, poi values used. Em,poi may vary with temperature.
The constants are set to the values required.
cm(3,3)
cl c2 0
c2 cl 0
0 0 c3
where,
cl = em / (1.0 - poi * poi)
c2 = cl * poi
c3 = cl * (1.0 - poi) * 0.5
B.13 Subroutine ELAW2 fEM.POI,G.THK.CP^
This subroutine forms Isotropic stress-strain law
ignoring temperature dependence of elastic constants,
cp(5,5) = constitutive matrix for plate element
em = Young's modulus
poi = Poisson's ratio
g
shear modulus

206
thk = plate thickness
The plate thickness is assumed to be constant and is
taken out of the integration and entered here.
The constants for plate stress-strain matrix are.
cp(5,5)
cp2 0 0 0
cpl 000
0 cp3 0 0
0 0 cp4 0
0 00 cp4
where,
ea = thk * thk * thk / 12.0
dem = 1.0 - (poi * poi)
eed=em*ea/dem
cpl = eed
cp2 = poi * eed
cp3 = g * ea
cp4 = g * thk / 1.2
B.14 Subroutine FORMH(R.S.NODE.XY.H.DJ.NN)
This subroutine forms four to nine node shape functions
and their derivatives in rs-space then transforms them into
xy-space through the inverse of Jacobian matrix.
DIMENSION NODE(9),XY(2,9),H(3,9),D(2,2), N ( 2,9)
N(2,9) = Coordinates of nine nodes in local system.
r,s = Natural coordinates in local system.
NODE(9) = Node numbers defining element.

XY(2,9) = local x,y coordinates of nodes
H(3,9) = Array of shape functions.
h(l,i)=f(i) h (2, i) =f (i),x h(3,i) =f (i) ,y
dj = Determinant of Jacobian,
nn = Number of nodes defining element.
D(2,2)=Jacobian matrix for 2D, replaced by Inverse of J
The initial functions are evaluated at the current
integration point by calling the subroutine GD two times
with the coordinate r and s of the current integration
point, respectively. The subroutine GD provides the
numerical initial functions and their derivatives
corresponding to the r and s coordinates provided.
The evaluation of shape functions starts with four
bilinear shape functions for the four corner nodes. If node
nine exists, the shape functions for the four corner nodes
need to be modified because the values of these corner node
shape functions at node nine is one fourth. This must be
forced to zero and can be done using the shape function of
the node nine because this shape function has the value of
one at node nine and zero at all other nodes. Therefore the
modification is simple subtraction of one fourth of the
shape function at node nine from the each shape function for
the corner node.
If any of the center nodes on the edge of the
element(nodes five to eight) exists, the shape functions of
corner nodes need to be modified further because the values

at center of the edge is one half. This can be done by
subtracting one half of the shape functions for the adjacent
center nodes on the edge of the element from the each of the
shape function for the four corner nodes.
The results are stored in the array H(3,9). H(l,i) are
the numerical value of the shape function evaluated at the
current integration point. H(2,i) and H(3,i) is the
numerical values of the derivatives of current shape
function with respect to to local variable r and s,
respectively.
Once shape functions are evaluated, Jacobian matrix can
be calculated as follows.
x = sum(fjL*x¿)
x,r = sum(f^,r*x¿)
y = sum(fi*yi)
y,r = sumif^^yj.)
x = sum(f^*x¿)
x,s = sum(f^,s*x¿)
y = sum(fi*yi)
y,s = sum(fi,s*yi)
where sum(qty(i)) = sum of qty(i) over i=l,nn
f¿ = the first row of h(3,9) matrix
f^,r = the second row of h(3,9) matrix
f^,s = the third row of h(3,9) matrix
x¿ = the first row of xy(2,9) matrix
y¿ = the second row of xy(2,9) matrix

It is noted that x¿, y^ are not the actual global
coordinates of node i of the element but they are local
element geometry coordinates. These values can be found by
the dot product of local axis unit direction vector before
being mapped into the natural coordinates and the vector
from the local origin and the node under consideration.
[J]
x,r y,r
x,s y,s
The inversion of two by two Jacobian matrix can be done
as follows. The determinant of Jacobian matrix is found.
D(1,1) and D(2,2) are interchanged and the signs are changed
to the opposite for the terms D(l,2) and D(2,l). All these
values are then divided by the determinant of Jacobian
matrix.
The element connectivity can be checked using the
determinant of Jacobian matrix. The determinant must be
greater than zero for the properly connected elements. If
the determinant is less than or equal to zero, an error
message is written.
The partial differential of shape functions with
respect to global coordinates x and y can be calculated as
follows.
fi,x = (f-^rH^x) + (fi,s)(s,x)
where, f¿,r = h(2,i), r,x = invJ(l,l), f¿,s = h(3,i),
s,x = invJ(1,2)

210
fi»Y = (f-^rH^y) + (fifs)(s,y)
where f^,r = h(2,i), r,y = invJ(2,1), f^,s = h(3,i),
s,y = invJ(2,2)
where invJ is the inverse of the Jacobian matrix.
B.15 Subroutine GD(B.IB.G.D)
b = Coordinate of local r or s of the current
integration point.
ib = Coordinate of local r or s of the current node.
g = Contribution to shape function in the current
r or s direction.
d = Derivative of g with respect to r or s direction.
For bottom corner nodes,
G=(1.0-B)*0.5
D=-. 5
For center nodes along the edge of element,
G=1.0-B*B
D=-2.0*B
For top corner nodes,
G=(1.0+B)*0.5
D=. 5
B.16 Subroutine TRANS (NN.TT.FKL.FKG}
This subroutine performs local-global transformations
of element stiffness.
nn
= Number of nodes for the current shell element.

tt(3,3) = Transpose of transformation matrix.
Sometimes called as [A] matrix for the second
order tensor.
fkl(45,45) = Local shell element stiffness. This has a
size of 45 by 45. There are five dofs at
each node, three translational dofs and
two out-of-plane rotational dofs. The
maximum number of element nodes is nine.
And hence 45 local dofs.
fkg(54,54) = Global shell element stiffness. This has a
size of 54 by 54. In global coordinate
system, even one local rotational dof may
have three components in global coordinate
system and thus there are six global dofs
at each node. The maximum number of
element nodes is nine. And hence 54 local
dofs.
Is(18) = Location matrix to put local 45 by 45 into
global 54 by 54 stiffness matrix. 54 dofs has
been divided into 18 groups of three
orthogonal dofs and will be transformed in
blocks of the three dofs.
It(18) = Location matrix for transformation matrix for
the stiffness transformation in blocks of
three. For the three translational dofs we
need all the three rows of transformation
matrix. For the two out-of-plane rotational
dofs we need two rows of three by three
transformation matrix.
Stiffness transformation in blocks of 3 is done through
the subroutine TRIPL, expanding 45x45 local stiffness to
54x54 global stiffness. As the above calculation is done
only for lower half of the matrix, the upper part of
stiffness is filled up using symmetry.

B.17 Subroutine TRIPLE(LI.L2.M.N.K1.K2.TT.A.AA.NR1.NCI.
NR2,NC2^
This subroutine calculates the matrix triple product,
aa = t(transpose) * a * t in blocked form (3x3). As the
transformation matrix has been formed in its transpose, the
product becomes aa=tt*a*tt(transpose).
where,
a(nrl,ncl) = Local element stiffness matrix.
aa(nr2,nc2)= Global element stiffness matrix.
tt(3,3) = Transpose of transformation matrix.
tk(3,3) = Temporary array to carry the results of the
first multiplication tt(3,3)*corresponding
a(3,3) .
11, 12 = Indices of element stiffness matrix in local
system.
m,n = Indices of transpose of transformation matrix.
kl, k2 = Indices of element stiffness matrix in global
system.
nrl = Number of rows in "a”,
ncl = Number of columns in "a".
nr2 = Number of rows in "aa".
nc2 = Number of columns in "aa".
The first multiplication, tt*a is performed followed by
the second multiplication [tt*a]*[tt(transpose)].

213
B.18 Subroutine INIT13 (NJTS.NDF.RF.FACILE
This subroutine forms initial loadings for the shell
element. Self weight is also included in this subroutine.
The application factor of temperature loading FACT
and/or the factor of self weight FACS are recovered from the
array FACIL transferred from base program. The applied loads
are then calculated using the equivalent nodal loads for
temperature differential and self weight computed in
subroutine EVEL13 and placed in the initial force array RF.
These equivalent nodal loads will be added to the global
load vector in the base program.
B.19 Subroutine STAT13 (NDF.0.TIME.FACAL.FACIL.ALFA)
This subroutine is for state determination
calculations.
COMMON /STLDPT/
This common block will be used to find load
application factor in RINT13.
Q(NDF) = INCREMENTAL global nodal displacements
Therefore, if total stresses are needed, the
incremental displacements are to be added up
at the end of each iteration.
QDL(54) = INCREMENTAL element nodal displacements
FACAL(1) = total load application factor
FACIL(l) = incremental load application factor

214
ALFA = factor used in FACAL
tt = transpose of element transformation matrix
nd = number of global displacements at nodes
The Gauss point parameters for linear stress
interpolation are set up. The global incremental
displacements are then added up and transformed to form
total local displacements using 3x3 submatrix to remove
multiplications with zeros.
Stresses at integration points are then calculated as
follows. There are six stress components at each integration
point.
[stress(24)] = [ST(24,54)][g(54)]
The initial stresses due to temperature loading are
subtracted from the calculated total stresses. The stresses
at four corner nodes are calculated linear extrapolation
from the stresses evaluated at the integration points and
put back into stress array sig(24).
B.20 Subroutine RINT13 fNDF.O.VEL.FE.FD,TIME,FACAL.
FACIL,ALFA)
This subroutine calculates element forces in the global
coordinate system.
COMMON /STLDPT/NPNF,NPFF,NPTP,NPP4,NPDP,NETP,NESWNITSR,
NSPATT
npnf = Number of nodal force patterns.

215
npff = Number of follower force patterns.
nptp = Number of nodal temperature patterns.
npdp = Number of nodal displacement patterns.
netp = Number of temperature loadings through
element data.
news = Number of self weight loadings through
element data.
COMMON /WORK / RFTEM(54),fktem(54,54)
rftem(54) = Temporary array for element force
rotation.
fktem(54,54) = Temporary array for element
stiffness transformation.
Q(NDF) = NODAL DISPLACEMENTS.
VEL(NDF)= NODAL VELOCITY
FD(NDF) = DYNAMIC NODAL FORCE WHEN TIME > 0
FE(NDF) = ELASTO-PLASTIC NODAL FORCE WHEN TIME = 0
TT = transpose of transformation matrix.
The element forces in the global coordinate system can
be calculated as
[FE global]=[Ke global][Q global]
As the element stiffness is in global coordinate system
and stored in the array FKP(1485). The element forces can
then be recovered directly from the global nodal
displacements.
The load application factor for temperature and/or self
weight loading is calculated. FACT is a factor for
temperature loads. FACS is a factor for self weight. These

216
are calculated whichever loading is applied. The load
application can be identified by the numbers of load
patterns, i.e., if NETP or NESP is positive, then there is
temperature loading or self weight, respectively.
IF temperature load is applied, subtract initial
temperature loadings from element forces. Templd(54) is
formed in initl3. Negative value of actual templd(54) has
been calculated because the base program uses -templd(54) as
applied loads. The + sign for subtraction is due to this
fact.
The index for gravity can be used to identify if self
weight is included.
B.21 Subroutine OUTS13 (KPR.TIME^
This subroutine is to print the time history of the
current state from /INFEL/ including stresses, strains,
status information, etc.
If the current element is the first element, write the
heading for the element information data and element
information. Otherwise print element information directly.
If no element information is requested, skip this
subroutine.
From the second element, write the element information
only. This includes element number, node number and stresses
at the four corner nodes. These stresses have been linearly
extrapolated from the integration points.

B.22 Linear Shell Element Data Input Guide
(1) Control information
(1-
1) First
control
line
COLUMNS
NAME
DATA
1
- 5(1)
NGR
Element group number = 13
6
- io(i)
NELS
Number of shell elements
11
~ 15(1)
MFST
Element # of first shell element
16
- 25(F)
DKO
Initial stiffness damping factor
26
- 35(F)
DKT
Tangent stiffness damping factor
41
- 80(A)
Optional heading for shell element
(1.
2) Second
control
line
COLUMNS
NAME
DATA
1
- 5(1)
NMAT
Number of shell material properties
6
- 10(1)
IGRAVD
Direction of gravity
11
- 15(1)
ITMPLD
Type of temperature load
16
- 30(E)
ALPHA
Thermal expansion coefficient
31
- 40(F)
REFTEM
Reference temperature
41
- 45(1)
ITMOFF
Temperature load turn-off index
46
- 55(F)
WGT
Integration weight (default = 0.999)
Notes:
IGRAVD : direction of gravity
global X Y Z -X -Y -Z
igravd 1 2 3 -1 -2 -3
ITMPLD : index for temperature load
1 = input top and bottom temperatures only
2 = input temperatures for all the nodes
else = no temperature effects
no input for the AHPHA,REFTEM,ITMOFF
ITMOFF : index to turn off temperature effect
1 = in local x direction (^xx = 0.0)
2 = in local y direction (<6yy = 0.0)
(2) Material property data
NMAT sets of material properties
COLUMNS NAME DATA
1 - 5(1) MAT
6 - 15(F) E
16 - 25(F) POI
26 - 35(F) G
Shell material property number
Young's modulus
Poisson's ratio
Shear modulus [default=E/(2.0*(1+POI))]

218
36 - 45(F) WGHT Self weight per cubic foot
(3) Element data
NELS sets of element data
COLUMNS NAME DATA
1 - 5(1) NEL Element number
6 - 10(1) MAT Material property number
11 - 55(1) NODE(9) Node numbers (915, 0 for missing node)
56 - 65(F) THK Element thickness (default = 1.0)
66 - 75(1) IJG(2) Number of element to be generated
If ITMPLD is 1, add a second line to element data.
1 - 10(F) TMPTOP Temperature at top surface
11 - 20(F) TMPBOT Temterature at bottom surface
If ITMPLD is 2, add two lines to element data.
1 - 64(F) Temperature at top surface for nine
nodes (9F8.2)
1 - 64(F) Temperature at bottom surface for nine
nodes (9F8.2)
Notes:
NODE(9): element node numbers
7*
8*
9*
4*
5*
6*
1*
2*
3*
IJG(2) : number of elements to be generated in i- and
j- direction including the element specified,
(works only for nine node elements)
If no temperature effects are desired for specific
element, input the same values for TMPTOP and TMPBOT,
i.e., (REFTEM,REFTEM)

APPENDIX C
IMPLEMENTATION NONLINEAR SHELL ELEMENT
This appendix describes the implementation of a
incremental nonlinear finite shell element into ANSR-III
program. Only those features that are different from those
of a linear shell element in subroutines INEL13, STIF13,
NONSTF, STAT13, STRESS, RINT13 are dealt with. For detailed
equations and elements of matrices, refer to chapters four
and five.
C.l Subroutine INEL13fNJT.NDKOD.X.Y.Z.KEXEC)
The local coordinates rs9(2,9) and the weights wg9(9)
for the standard 3X3 Gauss integration were added to /INFGR/
for the calculation of stiffnesses, stresses, and internal
resisting forces.
For the solution of the incremental equilibrium
equation in total Lagrangian description, the total
displacements and the total displacement gradients of
previous iteration are required and thus stored in /INFEL/.
These are ut9(9) through ayyt9(9). Some arrays for the
explicit integration through the thickness are also added.
These are thklyr(lO) through dhc(10). Accordingly, the
219

length of element information block LSTAT was changed.
The control variables for nonlinear analysis and
layered element are
IL = Number of layers.
KLD = Large displacement nonlinearity index.
1 = on, 0 = off
KGM = Geometric nonlinearity index.
1 = on, 0 = off
KMAT = Material nonlinearity index (not used).
1 = on, 0 = off
KUL = Motion description index.
1 = Total Lagrangian formulation.
2 = Updated Lagrangian formulation(not used).
The coordinate for 3x3 integration is
aw = 0.774596669241483
bw = aw
The layer information that can be handled currently is
10 layers with different thicknesses and materials,
nlyr = Number of layers.
thklyr(lO) = Layer thickness.
matlyr(lO) = Layer material.

C.2 Subroutine EVEL13(NJT.NDKOD. X. Y, Z .NODGEN.
TNODE.ICNTR.INERR}
The element is divided into layers for explicit
integration over element thickness (-t/2 to t/2).
Integrations needed are
htop
htop
dz ,
z dz , and
hbot
hbot
w
ht
z
hbot
§P
z^ dz,
where, htop = The coordinate of top of the layer.
hbot = The coordinate of bottom of the layer.
Thus the coordinates of top and bottom of each layer are
required and these are calculated and stored in hh(ll).
Currently the coordinate of the center line is set to zero.
But this can be changed for arbitrary location along the
thickness direction if an input parameter is given.
The explicit integration over the thickness becomes
[htop - hbot], [0.5(htop2 - hbot2)], and [(1/3)(htop3 -
hbotJ)]. These are calculated and stored in dhs(10),
dhc(10). It is noted that [htop - hbot] is equal to the
layer thickness itself.
The stress-nodal displacement matrix [ST] cannot be
used as it changes continuously in incremental nonlinear
analysis. [ST] is defined as
{stress} = [ST]{nodal displacement}

222
C.3 SUBROUTINE STIF13(ISTEP.NDF.CDKO.CDKT.
FK. INDFK. ISTFC)
The plate stiffness matrix [D] is calculated by
performing integration of constitutive matrix [E] through
thickness layer by layer using the subroutines ELAW1 and
ELAW2.
The incremental nonliner element stiffness is
calculated by calling the subroutine NONSTF. The rotation of
the element stiffness and the storage scheme are identical
to those for linear shell element.
C.4 Subroutine NONSTF (dm.dp.ds.lvr)
The standard 3x3 Gauss quadrature is used for the
numerical integration of nonlinear element stiffness.
The incremental nonlinear element stiffness requires
the total displacement gradient (w,x) and (w,y) from the
previous iteration. These were saved in wxt9(9) and wyt9(9)
for each integration point in subroutine STRESS and are
recovered for use.
Only the lower triangular portion of element stiffness
is calculated in this subroutine.
The linear element stiffness is formed using
subroutines SHSTF1 and SHSTF2.

The nonlinear element stiffness is calculated in two
groups; the large displacement stiffness [KLD] and the
geometric stiffness [KGM]. The large displacement matrix is
evaluated using the three component matrices [KLD1], [KLD2],
and [KLD3]. These are
[KLD1] = [Bli^tDHBn-j]
[KLD2] = [Bni]T[D][Blj]
[KLD3] = [Bni]T[D][Bnj]
The geometric stiffness [KGM] is calculated using the
element stresses at current integration points as
[G]T[STRM][G]. The inplane element stresses Nx, Ny, and NXy
for the current integration point are recovered from STRSL
array to form [STRM]. [G] consists of only (fi,x) and
(fi,y)•
C.5 Subroutine STAT13 (NDF.O.TIME.FACAL.FACIL.ALFA)
The element stresses are recovered through subroutine
STRESS at integration points using the incremental
displacements and then extrapolated to element nodes for
analysis and design purposes.

The locations of integration points for stress
evaluation are
4
X
I
I * 4
8 * 8
X
I
I * 1
I
1
X
+ (y)
-7
X
A
* 7
i
(0,0)
* 9
X
* 5
3
X
I
* 3 |
I
>
*66
X
I
* 2 |
5
x
2
x
+ (x)
where,
* = Integration points,
x = Node number for formulation.
This can be done using the shape functions [82] or bilinear
extrapolation for the inplane and shear stresses and
biquadratic extrapolation for the bending stresses. Another
approach is the least squares fit.
C. 6 Subroutine STRESS (ndf. cr)
Element stress evaluation is done at integration points
but is not a numerical integration. Thus no weighting factor
is used. The arrays used are
strsl(9,8,10) = Layer stresses
at 9 integration points,
for 8 stress components, Nx, Ny,
Nxy, Mx, My,Mxy, Qx, Qy,
and for 10 layers.

225
elstri(8,9) = Incremental element stresses
of 8 components
at 9 integration points.
strnip(8;9) = Total element strains
of 8 components
at 9 integ points.
The global displacements are transformed to local
coordinates for stress recovery. The material properties are
recovered to form the constitutive law or the stress-strain
relationship through subroutines ELAW11 and ELAW22. For each
layer the constants for explicit integration are recovered
and stresses are calculated at integration points. Stresses
are usually discontinuous and less accurate if they are
recovered directly at nodes. 3x3 integrations are used for
inplane and bending stresses and 2x2 integration points are
used for shear stress evaluation.
Incremental strains are calculated as
{strain} = [Linearized B matrix](nodal displacements}
= [BLZ]{q}
As [Blz] is the function of (w,x) and (w,y), these
incremental quantities, wx9(9) and wy9(9), must be
calculated first for the current integration point.
(w,x)= [sum of (fi)(wi)],x = [sum of (fi,x)(wi)]
(w,y)= [sum of (fi)(wi)],y = [sum of (fi,y)(wi)]

226
(fi,x) and (fi,y) are element of h(3,9) and wi is the local
nodal displacements transformed from the global nodal
displacements. And these are added up to yield total
displacement gradients, wxt9(9) and wyt9(9).
Total strains, strnip(8.9), are calculated because some
failure criterion requires principal strains. These are not
needed currently but will be used for material nonlinearity.
Once incremental strain is evaluated, then incremental
stresses can be calculated through explicit integration
across thickness and then added up for the total stresses.
C.7 Subroutine RINT13 fNDF.0.VEL.FE.FD.TIME.FACAL.
FACIL,ALFA)
This subroutine calculates element forces in global
coordinates, FE, the numerical integration of [BLZ]T{stress}
over the area. The {stress} are generalized, i.e., the
integration along thickness has been performed. [BLZ] is
evaluated in the same procedures used for stress recovery.
The integration scheme is compatible with that of stress
recovery, too. A 3x3 integration for the internal resisting
forces from inplane and bending stresses and a 2x2 for those
from shear stresses were used.

227
C. 8 Nonlinear Shell Element Data Input Guide
(1) Control information
(1.
1)
First
control
line
COLUMNS
NAME
DATA
1
—
5(1)
NGR
Element group number = 13
6
-
10(1)
NELS
Number of shell elements
11
-
15(1)
MFST
Element # of first shell element
16
-
25(F)
DKO
Initial stiffness damping factor
26
41
-
35(F)
80(A)
DKT
Tangent stiffness damping factor
Optional heading for shell element
(1.
2)
Second
control
line
COLUMNS
NAME
DATA
1
—
5(1)
NMAT
Number of shell material properties
6
-
10(1)
IGRAVD
Direction of gravity
11
-
15(1)
ITMPLD
Type of temperature load
16
—
30(E)
ALPHA
Thermal expansion coefficient
31
-
40(F)
REFTEM
Reference temperature
41
-
45(1)
ITMOFF
Temperature load turn-off index
46
-
55(F)
WGT
Integration weight (default = 0.999999)
56
60(1)
ILYR
Index for layer analysis
1 = layers of same thickness and
material.
Use for no layer analysis.
2 = 10 layers of different
thicknesses and materials.
61
mm
65(1)
KLD
Index for large displacement analysis.
0 = no 1 = yes
66
70(1)
KGM
Index for geometric nonlinear analysis.
0 = no 1 = yes
Note: KLD and KGM must be unity(1) for 'geometric' nonlinear
analysis if it includes the effects of large
displacements and initial stresses as used by some
authors.
71 - 75(1) KMAT Index for material nonlinear analysis.
(Currently not used.)
Notes:
IGRAVD : direction of gravity
global X Y Z -X -Y -Z
igravd 1 2 3 -1 -2 -3

228
ITMPLD : index for temperature load
1 = input top and bottom temperatures only
2 = input temperatures for all the nodes
else = no temperature effects
no input for the ALPHA,REFTEM,ITMOFF
ITMOFF : index to turn off temperature effect
1 = in local x direction (¿xx = 0.0)
2 = in local y direction (yy = 0.0)
(2) Material property data
NMAT sets of
material properties
COLUMNS
NAME
DATA
1
- 5(1)
MAT
Shell material property number
6
- 15(F)
E
Young's modulus
16
- 25(F)
POI
Poisson's ratio
26
- 35(F)
G
Shear modulus [default=E/(2.0*(1+POI))]
36
- 45(F)
WGHT
Self weight per cubic foot
(3)
Layer data if ILYR = 1
1
- 5(1)
NLYR
Number of layers(Current max. = 10)
6
- 15(F)
THKLAY
Thickness of layer
16
- 20(1)
MATLAY
Material Index of layer
(4)
Layer data if ILYR = 2
1
" 5(1)
NLYR
Number of layers(Current max. = 10)
1
- 80(F)
THKLYR
Thicknesses of layers( one line)
1
- 80(F)
MATLYR
Material Properties of layers(one line)
(5)
Element data
NELS sets of
element
data
COLUMNS
NAME
DATA
1
- 5(1)
NEL
Element number
6
- 10(1)
MAT
Material property number
11
- 55(1)
NODE(9)
Node numbers (915, 0 for missing node)
56
- 65(F)
THK
Element thickness (default = 1.0)
66
- 75(1)
IJG(2)
Number of element to be generated
If
ITMPLD is
1, add
a second line to element data.
1
- 10(F)
TMPTOP
Temperature at top surface
11
- 20(F)
TMPBOT
Temperature at bottom surface

229
If ITMPLD is 2,
1 - 64(F)
1 - 64(F)
add two lines to element data.
Temperature at top surface for nine
nodes (9F8.2)
Temperature at bottom surface for nine
nodes (9F8.2)
Notes:
NODE(9): element node numbers
7*
8*
9*
4*
5*
6*
1*
2*
3*
IJG(2) : number of elements to be generated in i- and
j- direction including the element specified,
(works only for nine node elements)
If no temperature effects are desired for specific
element, input the same values for TMPTOP and TMPBOT,
i.e., (REFTEM,REFTEM)

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233
31. Kamoulakos, A., "Understanding and Improving the
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32. Rhiu, J. J., Lee, S. W., "A Nine Node Finite Element
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33. Briassoulis, D., "Machine Locking of Degenerated Thin
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34. Yeom, C. H., Lee, W., "An Assumed Strain Finite Element
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35. Pinsky, P. M., Jasti, R. V., "A mixed Finite Element
Formulation for Reissner-Mindlin Plates Based on the
Use of Bubble Bubble Functions," I.J. for Numerical
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36. Pian, T. H. H., Chen, D., "On the Suppression of Zero
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in Engineering, Vol. 19. 1983, pp. 1741-1752.
37. Choi, C. K., Kim, S. H., "Coupled Use of Reduced
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38. Chan, H. C., Chung, W. C., "Geometrically Nonlinear
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41. Briassoulis, D., "The Zero Energy Modes Problem of the
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52.

53. Shames, I. H. and Dym, C. L., Energy and Finite Element
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joints in concrete box girder bridges," Master Thesis,
University of Florida, 1986.
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under Biaxial Stresses," ASCE Journal of Engineering
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236
66. Mehlhorn, G., Kollegger, J., Keuser, M., Kolmar, W.,
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237
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SUPPLEMENTAL BIBLIOGRAPHY
Ghaboussi, J., and Wilson, E. L., and Isenberg, J.,
"Finite Element for Rock Joints and Interface," Journal
of the Soil Mechanics and Foundations Division.
Proceedings of ASCE, Vol. 99, No. SM 10. October, 1973,
p. 833.
Wunderlich, Stein, Bathe, Nonlinear Finite Element
Analysis in Structural Mechnics. Springer-Verlag, New
York, N.Y.,1981.
Fafard, M., Dhatt, G., Batoz, J. L., "A New Discrete
Kirchhoff Plate/Shell Element with Updated Procedures,"
Computers & Structures. Vol. 31, No. 4. 1989, pp. 591-
606.
Yuan, F. G., Miller, R. E., A New Element for Laminated
Composite Beams, Computers & Structures. Vol. 31. No.
5, 1989, pp. 737-745.
Liao, C. L., Reddy, J. N., Engelstad, S. P., "A solid-
shell Transition Element for Geometrically Non-linear
Analysis of Laminated Composite Structures," I.J. for
Numerical Methods in Engineering, Vol. 26. 1988, pp.
1843-1854.
Ortiz, M., Morris, G. R., "C° Finite Element
Discretization of Kirchhoff's Equations of Thin Plate
Bending," I.J. for Numerical Methods in Engineering.
Vol. 26. 1988, pp. 1551-1566.
Kamoulakos, A., "Understanding and Improving the
Reduced Integration of Mindlin Shell Elements," I.J.
for Numerical Methods in Engineering, Vol. 26. 1988,
pp. 2009-2029.
Dhatt, G., Marcotte, L., Matte, Y, "A New Triangular
Discrete Kirchhoff Plate/Shell Element," I.J. for
Numerical Methods in Engineering, Vol. 23. 1986, pp.
453-470.
Hughes, T. Jr., Hinton, E., Finite Element Methods for
Plate and Shell Structures. Vol. l. Pineridge Press
International, Swansea, U.K., 1986.
Hughes, T. Jr., Hinton, E., Finite Element Methods for
Plate and Shell Structures. Vol. 2. Pineridge Press
International, Swansea, U.K., 1986.
238

239
Yang, T. Y., Saigal, S., "A curved Quadrilateral
Element for Static Analysis of Shells with Geometric
and Material Nonlinearities," I.J. for Numerical
Methods in Engineering, Vol. 21. 1985, pp. 617-635.
Chao, W. C., Reddy, J. N., "Analysis of Laminated
Composite Shells Using a Degenerated 3-D Element," I.J.
for Numerical Methods in Engineering, Vol. 20. 1984,
pp. 1991-2007.
Oliver, J., Onate, E., "A Toatal Lagrangian Formulation
for the Geometrically Nonlinear Analysis of Structures
Using Finite Elements. Part I. Two-Dimensional
Problems: Shell and Plate Structures," I.J. for
Numerical Methods in Engineering, Vol. 20. 1984, pp.
2253-2281.
Spilker, R. L., "Hybrid Stress Eight Node Elements for
Thin and Thick Multilayer Laminated Plates," I.J. for
Numerical Methods in Engineering, Vol. 18. 1982, pp.
801-828.
Noor, A. K., "Mixed Models and Reduced/Selective
Integration Displacement Models for Nonlinear Shell
Analysis," I.J. for Numerical Methods in Engineering.
Vol. 18. 1982, pp. 1429-1454.
Sander, G., Idelsohn, S., "A Family of Conforming
Finite Elements for Deep Shell Analysis," I.J. for
Numerical Methods in Engineering, Vol. 18. 1982, pp.
363-380.
Hughes, T. J. R., Liu, W. K., "Nonlinear Finite
Analysis of Shells-Part II. Two-Dimensional Shells,"
Computer Methods in Applied Mechanics and Engineering.
1981, pp. 167-181.
Hinton, E., Owen, R., Computational Modelling of
Reinforced Concrete Structures. Pineridge Press
International, Swansea, U.K., 1986.
Arnesen, A., Sorensen, S. I., Bergan, P. G., "Nonlinear
analysis of Reinforced Concrete," Computers &
Structures. Vol. 12. 1980, pp. 571-579.
Nilson, A. H., Design of Prestressed Concrete. John
Wiley & Sons, New YOrk, 1987.

BIOGRAPHICAL SKETCH
The author was born in Ham-ahn, Korea, in 1956. He
graduated from Busan Senior High School in 1974 and from
Seoul National University in 1978 with a B.S. degree in
architectural engineering. He then joined Korea Electric
Power Corporation and was involved in the construction of
nuclear power plants for 7 years.
The author received his M.E. degree in construction
engineering and management from the University of Florida in
1986. He expects to obtain his Ph.D. degree in structural
engineering in 1990.
240

I certify that I have read tluis study and that in my
opinion it conforms to /acceptably standards of scholarly
presentation and is fulw adequaty in scope and quality, as
a dissertation for the/^egree ofyDoctor of Philosophy.
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
liffüFd 0. Hays/ Jr./
Cliffy
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Fernando E. Fagundo^
Associate Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Jphn M. Lybas
Associate Professor of
ivil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Pagi W. Chun
jfessor of Biochemistry and Molecular Biology

This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August 1990
Dean,
lips
of Engineering
Madelyn M. Lockhart
Dean, Graduate School

07/02/2008 05:12 AM
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Nonlinear gap and Mindlin shell elements for the analysis of concrete

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1990
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Gainesville, FL 32611-7007




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xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0009018800001datestamp 2009-03-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Nonlinear gap and Mindlin shell elements for the analysis of concrete structures dc:creator Ahn, Kookjoon,dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00090188&v=00001001583924 (alephbibnum)23011869 (oclc)dc:source University of Florida