PERPUSTAKAAN UTHM

11111111111111111111111111111111111111111111111111111111111111111111111111111111 *30000001957478*

A study on non-friction contact problem with

large deformational analyses

Graduate School of Science and Engineering

Department of Civil Engineering

Saga University 2005

(Master Course)

04537025

Muhammad Nizam bin Zakaria

A study on non-friction contact problem with large deformational analyses

04537025 Muhammad Nizam bin Zakaria

1. Introduction When we analyze finite element structures with extremely large defonnation, non-linear problems are caused by not

only geometrical characteristics of an element but also elements' or nodes' contact phenomena. Contact problem is a

complex non-linear case and it is important to consider how to express the phenomena using digitalized data TIlere are

numerous types of contact phenomena that are adoptable in finite element analyses. For example, node-node contact,

node-element contact, node-surface contact, etc. From geometricaIly non-linearity of an element and the limitation of

segment element's size make the contact problem is extremely difficult to solve.

TIle main objective of this study is to apply Tangent Stiffness Method for simulation of a non-friction contact

phenomenon between nodes and elements. Furthennore, by adding some alteration to the contact judgment, various

types of structures can be applied to perfonn contact analysis. 1 A:aJ~si: Method

j Fig 1. Element Force and Global Coordinate

System for contact phenomena

N

V )

Fig 2. Nodal Forces for contact node and both ends

Here, derivation of calculation procedure for contact problem will be explained. Fig. 1 represents Element Force for

contact element in Global Coordinate System. Fig. 2 represents Nodal Forces for contact node and both element ends.

The relation between Element Force vector and Nodal Force vector in Equilibrium Equation can be expressed as

-a -I!.. -I!.. I!.. 1 u, 1 }

1~

Z,

u j

v} z) u,

-fJ

0

a

o o

a 1 1 fJ

a 1

o o

o 0

a -~I 1 1 } 0 0 fJ I!.. 1

1 ' a _~I 1 I'

,~, j AI) r

1 0 o -fJ o a

I)--i~

-------~

Eq. (I)

{a fJ}= Cosine vectors between both ends

1= Contact element length

I, = Length between end and contact

1 = Length between j end and contact )

Fig. 3 Element deformation

by Contact Force

node

node

Tangent Geometrical Stiffness for contact case is obtained by differentiating the matrix element in Equilibrium

Equation (Eq. (1». From the expansion and combination of Principle of Superposition. Element Force Equation for

contact problem is defined in Eq. (2).

0 0 0 0

_ 1'0/,0 EA 0 0 0 10 = on-stressed length for beam

0 (':00 J [~} [~1

I 2 1'0 I ',0 = cd length for i end and c node 0 4EI 2EI on- lr

3 Ell - -1'0' ,0 c: r 0 0 ' )0 = 0 + I , on- tr ed length fori end and c node

1,0/,0 --,-2-1)0 0

2EI 4EI 0 0 - -

C,o~,J , ,

0 0 0 0 0 Eq. (2) 1'0 ')0

3. Procedure for contact judgment

r" c • 0 ' ,,. ii ..... ~ 1;1:.." Ii 6J> c: c.

Ii /~ 0, ~ ii ~ ~ii h .. . J

T c I, I, I, I,

I

Fig. 4 Pre-contact situation Fig. 5 Po t-contact i tuati on

The contact judgment is perfonned by the combination of inner and outer product of vector values based on node i,j and

target node c. The cosine vectors for node i and target node cis Q nodej and target node cis b and cosine vector for

node i andj is c.

Fig. 6 Element deformation by Cootact Force Fig. 7 Definition ofrelea e contact

For release contact case, the judgment is perfonned when the value of Contact Force becomes les than zero. As

illustrated in Fig. 6 and Fig. 7, the perpendicular vector for cosine vector of the contact element c is defined as ii . When the Contact Force Y value becomes Ie than zero or changes direction, target node c is released and the

element will be defined as a non-contact element.

4. Analysis Results Case 1 Cantilever plane beam contact analysis

Young's Modules : 2.1 x1011

Area of cro section

Moment lnertia

Moment Force

t .

: 0.0050

: 0.0010

: 29.0364x 103

. .-- contact node

[N/m2]

[m2]

[m4]

[Nm]

20

2 3 .& , 6 7 8 9 lU II l2 13 14 IS 16 11 18

I .. I, "oOC).

Fig. 8 x.perimental model

.'")

... .--. .. . ~ ........ ~ ' .. '" ... ~

~ lep 30

Af, = 871 0898x lO) I ml

0, = 00157 Iradl

lep 51

M, = 1 219 - 10" 1 ml

°1 = 0.0274 Irad)

lep 141

HI = 68124 10' ['<ml

0, = 0122 Irlld)

.~ .. )

.. ..- '. +--.... ~ ••• " z

'i tep 142 te p 160

M, = 6.8764 xI06 [N ml M, = 80527" 106 1 ml

lep I 0

6662 10· 1 ml

01596 Irlld l B, = 0. 1144 Iradl B, = 0. 1353 Iradl B, =

Fig. 9 Element deformation for cantilever plane beam

Convergence of Maximum Unbalanced Force (Increment Step 160)

10000000000

~ ~ 10000000 :> 0

~ ~ 10000 E u ~ i 10 .. cOOl :l1:::J

0.00001 1 5 9 13 17 21 25 29

Iteration Steps

Unbalanced Force (N)

• Unbalanced Tor"",,, (Nm)

Fig. 9 represents contact phenomena for node and element fI r

cantilever plane beam element ep 142 repre n the

combining proce for contact element and the next element

that the contact node i about to pass trough. Thi proc

performed due to the increment tep of iteration number and

the Unbalanced Force valu i not po ible to con erge. Fig.

10 represen the convergent process of nbalan ed orce

vector in Step 160 in canti le er plane beam conta t case.

Fig. 10 Convergent process of Unbalanced Force (Step 160)

Case 3 Plane beam contact analysis (compulsory displacement)

I 2 J 4 5 6 7 8 9 10 II 12 13 14 I S 16 17 18 19

20

2J50m + 16 S0m

18 a 1,. 4000",

Fig. 11 xperimental model . . . . . . . . . . ... . . . ~ . . . - teplOI "

ontact node position contact node

(u20 , v20 ) = (23.5000m, -0.0 1 OOm)

. . .. , .... '. lep 400

Contact node position

(u20 , V20 ) = (23.5000m, -3.0000m)

Analy i condition

Young's Module

Area of cro section

Moment Inertia

: 2.1 x lO ll IN/m21 :0.0050 Im21

: 0.00101m41

ompulsorv di placement value: -0.0010 Iml

~ .. ........ .... , . .. .. ..

lep 200

onta t node position

(u 20 , v20 ) = (23.5000m, -1.OOOOm)

. . lep 444

ontact node po ition

(U20' v20 ) = (23 .5000m, -3.4400m)

• •

Fig. 12 lement deformations for plane beam

Convergence of MaXimum Unbalanced Force (Increment Step (44)

100000 10000

~ 1~~ ~ 10 ." I ~ 01 i 001

~ o~~~: :::J 000001

0000001 00000001

Unbalanced Force (N)

• Unbalanced Tor~. (Nm)

iteralton Steps

Fig. 13 on ergent pro e of nbalanced Force ( te p 444)

Fig 12 repre en contact phenomena for node and

element for plane beam element ep 444 represen

the combining process for contact element and th next

element that the contact node i about to p trough.

Thi pr i performed -wh n the length berneen

element end and the contact node becom than

3n from the length of the other end. Fig. 13 rep

the con ergent proc of nbalan ed Force \-ector in

ep 444 in plane beam contact case.

Case 8 Three rigid frames contact phenomena analysis (compulsory displacement)

~ .. o " .. -" 0 ~~ .. e " " c: e.!!!

. ~ 1l ~ c:

::lO::J

9 10

6 7

,

~ ~

20 27 ZI Young Modules :2. 1x I0" [ 1m2]

:0.0384 [m2l ~ ,.. 2S

II 37

20 21 22

8 " I I 1\

5 I' IS 16 31

~ ~~ ~ ~~

Fig. 14 Experimental model

}8 J9

" " n )J

~~

T r Area ofCro section

Moment Inertia : L.47 x 10-3[m~1

Value ofCompul ory Displacement : 0.0800 rm]

Model A: II nodes 14 e1emen ,

Model B: 17 node 24 elements

Model C:ll nodes 14 elements

t Dode

.

t-

,

-,

. . t .. l--

Step 39 Contact node position

u40 = 0.1200[m]

v40 = -31.0000[m]

~

. I

\ . .. . 'r l' . •

,<.,,,,,,,,,,,,~Z'..,,,,,,\..,,{{'...\..,,~,,\..,,,,,\..~1~\"',,,,,,,,,,~'W Step 60

Contact node position

u40 = 1.8000[mJ

v40 = -31.0000[mJ

\

Step 85 ontact node po ition

u40 = 3.8000[ml

v40 = -31.0000[mJ

\; -~7 , 1 I f

\ ,

Step 100 Contact node position

u40 = S.OOOO[m]

v40 = -31.0000[m]

Step 150 Contact node position

u40 = 9.0000[m)

v40 = -31.0000[mJ

Fig. 15 Element deformations for three rigid frames

tep 175 ontact node position

u40 = 11.0000 lmi

v40 = -31.00001ml

Fig. 15 repre ents the contact performance of three models. Convergence of Maximum Unbalanced Force (Increment

Step 100) With the analysis condition described above, compul ory

displacement is applied on node 40. Step 100 represents the

beginning step for contact between three models. As hown in

Step 175, model Chows the most deformation compare to

model A and model. B due to the increment value of Contact

Force between model B and model works on mod I . Fig.

100000 r,-:--.----

1000 Unbalanced Force (N) I

10 • Unbalanced Torque

(Nm) J

01

1 4 7 10 13 16 1 9 22 25

Iteration Steps 16 represents the convergent process of Unbalanced Force

Fig. 16 Convergent process of nbalanced Force ( tep 160) vector in Step 100 in this case.

4. Conclusion 1) Tangent Stiffness Method is able to solve this complex non-linear contact problem.

2) Proposed technique can express that the contact node lides 0 er elements end, thus providing strict convergence

results for the contact analy i .

3) Unbalanced Force con erges steadily, but in some cases, convergence process becomes worse, because of the

in tability of the node condition.

Contents

Introduction

Chapter One Large Deformational Analysis Theory

1.1 Tangent Stiffness Method for two dimensional structure analysis

1.2

1.3

1.l.l Outline

1.1.2 General equation of Tangent Stiffness Equation

1.1.3 Derivation of Ta ngent Geometrical Stiffness Ma trix

with energy principles

1.1.4 Iteration procedure for Tangent Stiffness Method

Application of Tangen t Geometrical Stiffness in plane frame structure

Definition for element behavior and integration of convergent solution

by Element Force Equation

Page

.. ·1

···3

···3

···3

···3

···4

···7

···11

···13

Chapter Two Contact Problem in a plane frame structure with large deformation ···17

2.1 Introduction ···17

2.2 Tangent Geometrical Stiffness for contact phenomena .. ·18

2.3 Definition for contact element behavior for contact problem

Chapter Three Programming

Chapter Four Numerical Examples

Case 1 Cantilever plane beam contact analysis

A Numerical experiment to check the limit of sliding

over element end

B Modification to allow sliding over element end

4.1 Definition of sliding over element end

Case 2

Case 3

Consideration for Case 1 A and B

Cantilever plane beam contact analysis (compulsory displacement)

Consideration for Case 2

Plane beam contact analysis (compulsory displacement)

Consideration for Case 3

···23

···27

···30

···30

···37

···39

···43

···44

···51

···52

···60

Case 4 Fully constrained column contact analysis with reversible

compulsory displacement to define contact release

4.2 Definition of contact release

Consideration for Case 4

Case 5 Two cantilever plane beams contact analysis

(compulsory displacement)

Case 6

Case 7

Case 8

Case 9

Consideration for Case 5

Two rigid frames contact phenomena analysis

(compulsory displacement)

Consideration for Case 6

Three rigid frames contact analysis

(compulsory displacement)

Consideration for Case 7

Three rigid frames contact phenomena analysis

(compulsory displacement)

Consideration for Case 8

Three rigid frames contact phenomena analysis

(compulsory displacement)

Consideration for Case 9

Chapter Five Conclusion

Few words of thanks

References

2

Page

.. ·61

···63

'''71

···72

"·79

.. ·80

.. ·89

· .. 90

.. ·97

.. ·98

"'106

"'107

"·114

.. ·115

"'117

'''118

Chapter One Large Deformational Analysis Theory

1.1 Tangent Stiffness Method for two dimensional structure analysis

1.1.1 Outline

This chapter presents the derivation of Tangent Stiffness and Tangent Geometric Stiffness

equations; also the chapter describes iteration process for Unbalanced Force equation solution and

geometrical non-linear structure analysis technique for Tangent Stiffness Method.

The Tangent Stiffness Equation can be easily derived by differentiation of Equilibrium

Equation that connects Nodal Force vector in global coordinate system and Element Force vector

in local coordinate system. Element Force that works on element end is completely prescribed by

Element Force Equation. In this equation, element stiffness is independent from the element

displacement; it is caused by the non-linearity of the Tangent Geometrical Stiffness. Also the

Element Force Equation precisely represents element behavior, avoiding approximation even for

complex cases.

Furthermore, by applying Principle of Stationary Potential Energy, it is possible to express a

symmetric matrix for Tangent Geometric Stiffness. Using this Tangent Geometric Stiffness

matrix makes possible to express strict element behavior that prescribed in the Element Force

Equation.

Tangent Stiffness Method avoids the derivation of complicated non-linear element stiffness

equation with non-linear material properties, which is quite difficult to formulate. Only strict

Compatibility Equation and Equilibrium Equation are used for iteration process to converge

the Unbalanced Force.

Iterative solution in Tangent Stiffness Method is mathematically equal to Newton-Raphson

Method technique. Comparing the Tangent Stiffness Method to the Newton-Raphson Method

applied in Finite Element Method (FEM), it shows overwhelmingly high efficiency of Tangent

Stiffness Method in convergence performance.

1.1.2 General equation of Tangent Stiffness Equation

In one element inside a finite element structure, Element Force vector S and element

deformation vector s related to Element Force Equation are defined as in the equation (1.2),

(disintegration is completed at this point).

S=ks (1.1)

( k :=: Stiffness Matrix)

After differentiation, it becomes

os = Kbs (1.2)

3

The tangent line of element force equation is applied only in case of considering geometrical

non-linear performance of element deformation component. In addition, if the definition of

element behavior is linear, then element stiffness is defined as k = K .

Here, if local coordinate system of one element component shows nodal force vector in a primary

balance condition is D, Equilibrium matrix is J, the next Equilibrium Equation is derived.

JS=D (1.3)

After differentiation,

(1.4)

where 8S and 8.J, make possible to strictly express the linear function of nodal displacement

vector &I in the local coordinate system (1.4), it can be rewritten as follows

(1.5)

. The equation (1.5) shows the Tangent Stiffness Equation for Tangent Stiffness Method. Here

Ko is Element Stiffness Matrix, obtained by converting K of equation (1.2) into a local

coordinate system in Compatibility Equation which is calculated at each iteration step. KG is a

Tangent Geometrical Stiffness with non-linear characteristics from Compatibility Equation

which links nodal displacement vector and element deformation vector. It is also essential to

develop an equation that strictly connects the geometrical non-linear characteristics and rigid body

displacements. In Tangent Stiffness Method, strict Tangent Stiffness Equation can be obtained

by a concise induction process without calculating non-linear stiffness equation. For this, the

complexity of the induction process in Lagrangian style Finite Element Method is relatively

complicated comparing to the method mentioned above.

1.1.3 Derivation of Tangent Geometrical Stiffness Matrix with energy principles

Regarding to equation (1.4) and equation (1.5), if Tangent Geometrical Stiffness KG is

expressed as

K _ 8(JJS) G - 8Jd

(1.6)

according to expansion of Principle of Stationary Potential Energy, an element force vector

obtained from primary balance condition leads into expression of Tangent Geometrical Stiffness

matrix.

4

Nodal Force (External Force)

Equilibrium Equation Stiffness Equation

D

Force Nodal

Displacement

u

Lis

Element Force Equation Compatibility Equation

Element Deformation

Figure 1.1 Energy - vectors relationship in Tangent Stiffness Method

5

Figure 1.1 shows nodal displacement vector, element deformation vector, Element Force vector.

and Nodal Force vector, expressed as scalar quantities versus energy. The first quadrant represents

Stiffness Equation, the second quadrant - Equilibrium Equation, the third quadrant - Element

Force Equation, and the last quadrant - Compatibility Equation. In addition, the inner rectangle

shows primary balance condition, and the outer rectangle presents balance condition after nodal

force increment when external force has been applied. Here, if Strain Energy is V and

External Energy is U ,so, the Total Potential Energy II is

lI=U-V (1. 7)

After deformation, Total Potential Energy II' can be expressed in the following way

(1.8)

If balance condition before and after deformation is assumed to be constant, according to the

Principle of Stationary Potential Energy,

81I =0 8Ad

81I' = 0 8Ad

(1.9), (1.10)

Figure 1.1 shows that V; is not being influenced by the increment of nodal displacement Ad ,

therefore,

(1.11)

The differential for increment of nodal displacement Ad for Vii, AV ,U li ,and

AU becomes

8AV =AD 8Ad

8U# 8AsT -=--S 8Ad 8Ad

8A U = _8_ f" ASdAs = 8L1sT ~ t' ASdAs = 8AsT

AS ~d ~d ~d~s ~d

So, equation (1.10) can be rewritten as follows.

8AsT

(S+AS)=D+AD 8Ad

Therefore, it is clear that equation (1.16) and equation (1.3) are identical.

J +AJ = 8AsT

8Ad

(1.12), (1.13), (1.14)

(US)

(U6)

(I.l7)

According to equation (1.4) and equation (1.5), the Tangent Geometrical Stiffness is

6

(1.18)

If a finite increment of element deformation vector As expands as a quadratic function Ad , the

work quantity that performed by element force S and increment As, the Tangent Geometrical

Stiffness KG can easily be expressed by second differential of Ad ,according to geometrical

quantity and dynamic quantity in primary balance condition. Therefore, the Tangent Geometrical

Stiffness Matrix from equation (1.18) expressed in particular form is

(1.19)

1.1.4 Iteration procedure for Tangent Stiffness Method

The procedure of Tangent Stiffness Method can be implemented in the way described below. The

iteration technique proceeds until convergence of Unbalanced Force to a strict balance position by

using loading steps. The common procedure steps are:

Primary displacement

Primary load

Load increment

Displacement for step of iteration (r)

Element deformation vector

Element Force vector

Element Force - Nodal Force Conversion Matrix

Tangent Stiffness Matrix

Ado

Do AD

Adr

Ado

Do J r

K Tr

Figure 1.2 shows a convergence concept diagram for Tangent Stiffness Method. The

calculation goes clockwise. The first quadrant represents relation of displacement and load. In

Tangent Stiffness Method, non-linear stiffness equation is not involved in calculation process

and it is marked as a dotted line on the graph. The fourth quadrant represents the condition of

compatibility, that expresses relation between nodal displacements Ad in global coordinate

system and element deformation vector As in the local coordinate system. The third quadrant

represents Element Force Equation; where element behavior is prescribed in order to obtain

element force vector S from element deformation vector. The second quadrant represents

Equilibrium Equation, which is obtained from Element Force vector and the coordinate

transformation for the current displacement. This is necessary to calculate in order to seek Nodal

7

Equilibrium Equation

I

I I

I I

"

I

" " "

I I

Stiffness Equation

,'/F

I I

~S+-~-+---TJ~~D~--r-~B=+~H~ __ ~~--+~d

J

Element Force Equation Compatibility Equation

~s

Figure 1.2 Iteration concept of Tangent Stiffness Method

Force vector.

In Figure 1.2, iteration process in Tangent Stiffness Method begins from given state of balance,

which starts from the point 0, and the following steps explain the iteration solution process until

convergence.

CD c5d1 is obtained by solving Tangent Stiffness Equation at point 0, for given value of load

increment AD. (O-A-B)

8

® calculate Ad] by adding od] to displacement Ado in primary state.

@ from displacement Ad] obtain strict compatibility equation that is used to solve element

deformation vector As. (B~C)

@ from the Element Force Equation S] , the state of equilibrium, F is provided by

calculation for AD] using the Equilibrium Equation. (C~D~E~F)

® Unbalanced Force oD] is calculated by cyclic calculations, from the difference of load

condition AD and load vector AD] which is in balance state in displacement

position Ad].

@ in this stage, condition F is considered as the primary state, and it finds solution by Tangent

Stiffness Equation od2 which corresponds to oD]. (F~G~H)

({) New displacement for nodes Ad2 is calculated using the previous od2

@ calculation for the next state of balance L is performed in the similar way H~ I~J~ K.

® The rest of the steps is repeated continuously until the state of balance in first quadrant get s

closer to Z, and the convergence solution is obtained for this calculation procedure.

Summary, iteration process in Tangent Stiffness Method can be expressed as

(1.20)

Thus, from the comparison between iteration process of Finite Element Method, equation

(1.20) and calculation performed by Tangent Stiffness Method, it is not necessary to formulate or

apply any approximation concept. Although it is almost impossible to express strict solution using

non-linear stiffness equation in reality, (actually, it is quite impossible to be expressed with any

analytic technique, and furthermore, it is not necessary to express any of it in this method). The

results from the iterating process which was explained previously, is presented by the dotted line

in Figure 1.2 that shows non-linear stiffness equation which is solved strictly while passing

through the curved line in a random step.

In addition, Figure 1.3 is a flow chart for a geometrical non-linear structure analysis program

performed by Tangent Stiffness Method According to the figure, the expression for each

coefficient of Tangent Stiffness Matrix was obtained from the Unbalanced Force calculation; and

it shows that composition of a logical algorithm becomes possible without any complicated

procedure such as numerical value integration.

Furthermore, this method can be easily adapted for three-dimensional rigid frame analysis

which requires considering rotational displacements. As in this method, node rotation is defined as

independent from strain point in an element, and it is possible to apply rotational composition

technique by using coordinate transformation matrix.

9

Element characteristic value, Initial node coordinate

value, Element division terms, Boundary condition,

Load increment Ll D input

~ I Calculation of element force stiffness coefficient k n I

r = r+1

Calculation of Unbalanced Force

Element coordinate system settings

*Calculation of Element Deformation (element dimensions, rotate deformation)

Ll s by strict Compatibility Equation

Calculation of Element Force 5, from Element Force Equation

In case of linear Element Force Equation : 5, = KoLl s, In case of non-linear Element Force Equation: Newton-Raphson Method

Calculation for balancing the Unbalanced Force aD, = Do + Ll D - f,S,

Convergence check YES Converged J

loDrlmax(& NO

P"p~ation ofT~g.nt StiOh." Matrix '(2: )1 Tangent Geometrical Stiffness

. a Ll SkSk

. KG. = aLl daLl d I J <ld,.<ld, .... <ld.-->O

Tangent Element Stiffness : Ko = frkorfr T

K Tr = Ko +KG

Calculation of simultaneous linear equation

(Gauss Elimination Method)

&l = K -1 8D r Tr r

*Updating node coordinates

Ll dr+1 = Ll dr + ° dr

Figure 1.3 A basic flow of a structure analysis program performed by Tangent Stiffness Method

10

1.2 Application of Tangent Geometrical Stiffness Method in Plane Frame Structure

It is possible to obtain Tangent Geometrical Stiffness for plane frame structure by substituting

expansion of Compatibility Equation to equation (1.19). However in this chapter, derivation by a

simple induction process which requires differential of Equilibrium Equation will be performed.

r-----~ll

v

Figure 1.4 Element Force and Coordinate System for beam element

r-----~1I

v

~.

Figure 1.5 Nodal Force for both ends of an element

Figure 1.4 represents one element of a plane frame structure, when the support conditions are

set as stable and static, Element Force Equation for this combination of element edge forces.

corresponding to the support conditions is shown in the next equation.

(1.21)

11

Further, node i is a pin fixed node, and node j is movable in element axial direction or a roller

node, direction from node i to node j is set as primary axial direction, and beam coordinate system

is applied for element coordinate system. In addition, Figure 1.5 shows that when replacing global

coordinate system to element coordinate system, expression of Nodal Force that works on the

same element's both ends can be displayed in following vector form.

(1.22)

Therefore, if directional cosine vector is {a,,8} and element length is I , then Equilibrium

Equation between Element Force vector and Nodal Force vector can be expressed in matrix form.

-a ,8 ,8 U j I I

V; -,8 a a I I

[~:l Zj 0 1 0 (1.23) = ,8 ,8 U j a

V I I } a a

Zj ,8 I I

0 0 1

Here, if node coordinates for both ends are expressed as uij = uj - u, , vij = Vj - Vi ,differential

for each components of Equilibrium Matrix in equation (1.23) are shown as follows.

6( ~ ) = /2 {(,82 - a 2 }>Uij - 2a,86Vij}

6(~)= /2 {-2a,86Uij _(,82 -a2~'J

(1.24)

(1.25)

(1.26)

(1.27)

(l.28)

Tangent Geometrical Stiffness matrix KG is obtained in similar way in (1.31) by differentiating

Equilibrium Equation (see equation (1.23»; and Element Force is considered constant.

12

l p' -afJ 0] l2ap fJ2 _a2

~] kG ~ ~ -~p a 2 ~ +7 P'/ -2afJ 0 0

(1.29)

Q=M;+M j

1 (1.30)

K =[ kG G -k

G

-kG] kG (1.31)

In addition, by removing the rotation component, equation (1.29) is the same form as Geometrical

Stiffness for plane axial one dimensional element (truss element).

1.3 Definition for element behavior and integration of convergent solution by Element Force

Equation

Figure 1.6 shows non-stressed state for two nodes of a beam element which is defined as a

straight line, and it is considered in balance state when axial force N and edge moment M; ,

M j are applied on both ends of it. Here, Tensile Stiffness EA ,Bending Stiffness E1 ,

and non-stress length 10 are set for the beam. In addition, Us ---+ Vs coordinate is assumed as

the primary axis of element axial direction for the Local Coordinate System.

When considering small linear element of beam element in an equilibrium state, the balance of

moment on the right end is;

(1.32)

if there is no intermediate force, differential for Shear Force Q will be 0 and the differential for

equation (1.32);

2 d' d M +N -~ =0 2 d 2 du, 11.,

(1.33)

where, if we substitute M = -E1(d2vj dUs2

) to equation (1.33), then the differential equation

of deflection will be as follows.

13

~-----------------l------------------~

Figure 1.6 Element Force and element deformation quantity

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

v Q

N~---+-+

\4--------- du s -------.-:~

dM M +---du,

du" .

dv v +--=-du - d S

Us

dN -H----.L..~N + ---du du

s s

0+ dQ du - d S u,

Figure 1.7 Balance of a small linear element in a beam element

14

(1.34)

if we solve the differential equation according to the following boundary condition,

~(O)= 0 (1.35), (1.36)

dv =B;

dv =B -- ---+=-

dus dus )

x=o x=/o

(1.37), (1.38)

(1.39)

(1.40)

it can be displayed as shown in equation (1.39), and the relation of end moment force and

deflection angle are obtained. In addition, coefficient for deflection angle a or b, are defined by

the plus or minus of axial direction N .

N>O

0) cosh 0) - 0) sinh 0) a = ------;;----~ 0) sinh 0) + 2(1 - cosh 0) )

b = 0) sinh 0) - 0)2

0) sinh 0) + 2(1 - cosh 0))

0)2 cos 0) - 0) sin 0) a = -----,-------c-

0) sin 0) - 2(1- cosO))

• 2 b = 0) sm 0) - 0)

0) sin 0) - 2(1- cos 0) )

0) = lo~ N E1

Next, the matrix form expression for equation (1.39) can be expressed as

M=GO

(1.41), (1.42)

(1.43), (1.44)

(l.45)

(1.46)

By differentiating equation (1.46), general form for Tangent Element Force Equation can be

expressed as 5M = GOO + OGO

= G{5() + dG BON dN

= GOO + liON

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(1.47)

Further, differential of the coefficient for deflection angle can be expressed as

dG [p dN = q

a-b 2

P=--,-ro-

a+2b-ab q=

(1.48)

(1.49), (1.50)

As a conclusion, idealization and simplification of equation (1.47) shows that it is possible to

prescribe clement behavior. Hereby Element Force Equation and Tangent Element Force

Equation combination is possible to be defined in both cases for straight element and curved

clement, in order to apply it to this mathematical algorithm.

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Chapter 2 Non-friction contact problem in a plane frame structure with large deformations

2.1 Introduction

Contact phenomenon is complex nonlinear problem to solve which and it is important to

consider the physics of the problem such as load conditions, material properties, support

conditions in order to achieve a strict convergence solution. The description of contact between a

node and a beam element can be simply illustrated in the following figures. Figure 2.1 represents

pre-contact case and Figure 2.2 represents post-contact case. The existence of geometrical

material between node i and node j, prevents the target point (contact node) pass through the

beam element. The calculation for Element Force Equation in this case is almost the same for the

plane frame calculations that were explained in the previous chapter. The only difference is

modification of contact element by implanting additional node. The calculation procedure will be

explained in details in the following page.

c. ..--Target point

~N j

Figure 2.1 Pre-contact case

@ I 1

N

Figure 2.2 Post-contact case

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