I. OPERATION OF THE OPTION 2 fileII. BAR ELEMENTS IN THE NON-LINEAR ANALYSIS IN THE ROBOT MILLENNIUM...

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MANUALS

Elasto-Plastic Analysis

I. OPERATION OF THE OPTION 2

1. Assumptions 2

2. Calculation Methods 2

3. Management of the Calculation Process 6

4. Analysis of Results for Elasto-Plastic Bars 6

II. BAR ELEMENTS IN THE NON-LINEAR ANALYSIS IN THE ROBOT MILLENNIUM SYSTEM 8

1. Preliminary Remarks and Assumptions 8

2. Geometry, Kinematics and Strain Approximation 8

3. Stresses and Internal Forces within an Element 11

4. Geometrical Non-Linearity 16

5. Elasto-Plastic Hinges 19

6. References 20

III. VERIFICATION MANUAL 21

EX 1. Plastic Hinge in a Rectangular Beam 22

EX 2. Limit Moment Analysis 23

EX 3. Plastic Response to a Suddenly Applied Constant Force 25

EX 4. Plastic Bending of a Clamped I-Beam 26

EX 5. Two-Span Frame 28

EX 6. Slender Frame 30

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MANUALS

I. Operation of the Option

1. Assumptions

The linear structure analysis assumes elastic and linear material behavior (once loads are removed, values of deformations and stresses in a structure return to the initial state - before the load was applied and between stresses and deformations the linear relationship is assumed).

The elasto-plastic analysis of the structure in the Robot Millennium program enables considering material non-linearity. Once the yield strength is exceeded, material stiffness is different from that within the elastic range. Secondly, after exceeding the yield strength, deformations induced in the plastic state may remain after the unloading.

Non-linear material Plastic return path

However, it should be added here that in the assumed material non-linearity the change of material stiffness depending on external factors, such as temperature is not taken into account; rheological issues (change of material properties in time) are not included in considerations, either. The basic assumptions of the elasto-plastic analysis in the Robot program comprise as follows:

• the option works both for 2D structures (2D frames, grillages) and for 3D structures

• only normal stresses resulting from longitudinal forces and bending moments are considered (tangential stresses caused by transversal forces and a torsional moment are disregarded)

• elasto-plastic analysis is performed for selected bars indicated by the user; it is assumed that the analysis type is not changed globally for the entire structure since analysis of this type is time-consuming and requires definition of local conditions for a bar (division of a section or bar, material model)

• elasto-plastic analysis is performed only for structure bars – it is not performed for surface or

volumetric finite elements.

In the current program version the method adopted and the elements used impose certain restrictions concerning elasto-plastic analysis:

• material section is uniform,

• material model exhibits compression-tension symmetry and elasto-plastic behavior with hardening, thus practically, the analysis works for steel sections

• the analysis provides support for rolled sections from the section databases and for parametrized steel sections defined by the user,

• the analysis does not support bars with tapered section,

• the analysis does not support bars on elastic ground (Winkler’s ground),

2. Calculation Methods

The analysis of elasto-plastic bars is carried out on several levels: element -> section -> point (material model). For each of these levels different solutions of calculation methods may be adopted: Next chapters present a description of methods that have been adopted for the following problems: 1. Analysis on the element level– strategy of discretization of the elasto-plastic bar, 2. Analysis on the section level – calculation of stresses or forces in the section of an elasto-plastic

bar, 3. Material models – the relationship between stresses and deformations, unloading method.

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MANUALS

Analysis on the Element Level A structure model consists of finite elements. In the elastic analysis it is sufficient to calculate displacements and forces in structure nodes. Afterwards, forces and stresses in elements are approximated based on the nodal values and loads acting on the elements. In the elasto-plastic analysis forces and stresses should be examined directly within an element, since there is no possibility to predict the stresses over the element’s length. Of course, in numerical calculations certain division (discretization) of an element will be applied and the analysis for points along the element length will be performed. To the user this division is implicit. The user applies a bar, i.e., an element or elements with properties ascribed. In the elasto-plastic analysis, discretization within a level is conducted by ADDING GLOBAL DEGREES OF FREEDOM. Bars are divided into smaller calculation elements. Additional nodes and calculation elements are invisible to the user. A number of nodes and global degrees of freedom in a project is growing. In each of the elements, calculations of stresses are carried out at three points (third-order Gauss quadrature is applied here). Default division of elasto-plastic bars is applied automatically. A division method may be selected by using the Division of elements for elasto-plastic analysis option provided in the Job Preferences dialog box (non-linear analysis options). There the user may define a fixed division number for each bar or choose the automatic division method. The method of automatic division may be determined using the Maximum element length option located in the Analysis Type dialog box (the Structure Model tab). The user defines the maximal length of a calculation element. Next, the automatic division proceeds in such a way that, the minimal initial division of an element into 5 parts is assumed. If after dividing the total element length through the specified maximal length, a division number greater than 5 is obtained, then the obtained greater division number is adopted. Thus, each bar is divided to at least 5 calculation elements whose lengths do not exceed the value specified by the user.

Analysis on the Section Level Independently of the method of bar division over the length, the analysis involves verification of stresses at a given point on a bar. This point is ascribed a defined section. For the verification of stresses in a section the layered structure approach (division into zones) has been assumed.

A section is divided into a set of layers (fibers). In a bi-axially bent section they may be rather referred to as zones. In each of the zones stresses are being checked according to the assumed model. Forces acting on the bar axis are obtained by integration of forces in all section zones.

The figure beside illustrates an example of section division into zones. Let us assume that the number of zones equals N. The following parameters have to be determined for each of section division zones: - Coordinates (yi, zi) of the point being the zone’s

center of gravity in the main central coordinate system of the section,

- Area of zone A i, - Material M i assigned to the zone, ; where the i index denotes a zone number and i=1..N.

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MANUALS

The analysis is performed in such a manner that for each load increment, the program calculates displacement increments at division points along the bar length. Afterwards, based on the displacements, deformations at points in a section are computed. Considering the function that describes a material model for a given zone, stresses are calculated at each point depending on the current deformations. Next, on the basis of stresses internal forces are calculated. Finally, internal forces at all points (zones) should be added up (integrated) to obtain internal forces in a bar. Such an iteration procedure is being repeated until the full loading is achieved.

This option is described in EC3 (point 5.2.4.1(8)) as elasto-plastic analysis. It allows for bilinear strain-stress relation or more exact relationships. A critical section remains entirely elastic until the yield strength is reached at the extreme point. Further increase of stresses is accompanied by the gradual section ”yield” along with expansion of plastic zones down in the section’s depth and over the element length. This is the approach which reflects the physical phenomenon with great fidelity. It enables implementing composite sections and a user-defined material model. It requires, however, storing large amount of data due to section division into zones, while the increase of accuracy and convergence of iteration procedures calls for a large number of section division.

Material Models

In this case the bilinear material behavior is assumed, i.e., elastic and linear in the plastic range. A model is generated by means of data included in the material database, based on Young’s modulus (E) and yield strength (Re) of a selected material. There are two material models available:

Material: elastic-perfectly plastic Material: elasto-plastic with hardening

Value of hardening E1/E is determined by the user. Material models of this type are allowed for by EC3 (point 5.2.1.4 (8) and (9)).

In the plastic analysis the unloading method should be defined, as well. It determines the path of material behavior after exceeding the yield strength - during unloading, when deformations become smaller (deformation gradient is negative).

There are four ways of material behavior to select from (see the drawing below):

• elastic - return along the same path as while loading, in case of the non-linear material model, this will be the non-linear elasticity,

• plastic - return along the path defined by the initial modulus of elasticity, value of parameter a = 1

• damage - return along the path directed to the point (,) = (0,0); value of parameter a = 0,

• mixed - return along the intermediate path between the plastic path and damage path determined by the parameter a : 0< a < 1.

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MANUALS

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MANUALS

3. Management of the Calculation Process

In the standard non-linear analysis the load is applied in increments dQ = 1.0 / X, where X – a number of load increments; consequently, the maximum possible load factor (Q) which may be reached for convergent calculations, equals Qmax=1.0.

Additional criteria of stopping the analysis, provided in the dialog box available in the calculation parameters of a non-linear case, enable execution of the non-linear analysis in such a manner that the maximum load factor Qmax is defined by the user or determined by certain event. The following criteria which allow stopping the analysis are available in the dialog box: 1A Structure collapse 1B Reaching load factor 2A Plasticity state (beginning of the plasticity state at any point of the section) 2B Reaching plasticity rate 3A Exceeded maximum displacement of any node 3B Exceeded maximum displacement of a selected node Let us explain the above definitions of the analysis stop criteria:

• Criterion - Structure collapse The analysis is performed until the divergence of the iterative process is reached.

• Criterion - Reaching load factor The analysis is performed until the value Qmax = Qdef (where Qdef is a load factor determined by the user) is obtained or until the divergence of the iterative process is reached; the analysis is stopped, if before one of the conditions mentioned occurs.

• Criterion - Plasticity state The analysis is performed until the plasticity process is started at any point of the section of at least one elasto-plastic element or until the divergence of the iterative process is obtained; the analysis is stopped, if before one of the conditions mentioned occurs.

• Criterion - Reaching plasticity rate The analysis is performed until the plasticity process of any elasto-plastic element reaches the limit value of the plasticity state determined by the user or until the divergence of the iterative process is obtained; the analysis is stopped, if before one of the conditions mentioned occurs.

• Criterion - Exceeded maximum displacement of any node The analysis is performed until the value of maximum displacement of any node exceeds allowable displacement determined by the user or until the divergence of the iterative process is reached; the analysis is stopped, if before one of the conditions mentioned occurs.

• Criterion - Exceeded maximum displacement of a selected node The analysis is performed until the value of maximum displacement of a selected node exceeds allowable displacement determined by the user or until the divergence of the iterative process is reached; the analysis is stopped, if before one of the conditions mentioned occurs. There is a possibility to define more than one criterion of stopping the analysis; however, the definition is limited to one criterion from a given group, i.e. 1, 2 or 3. Thus the user may select for example 1A, 2A, 3B, but the definition 1A, 1B or 2A, 2B or 3A, 3B is not allowed.

4. Analysis of Results for Elasto-Plastic Bars

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MANUALS

For bars with elasto-plastic sections there are identical results available as for bars with standard sections. The user may obtain deflections, internal forces and stresses at any point along the bar length. Afterwards, elasto-plastic members may be verified and designed in the program by means of calculations for steel codes.

Apart from that, the section plastic coefficient is available for bars with elasto-plastic section. It is the ratio of the area of plasticized section part to the total section area. It assumes values from 0.0 – for an elastic section to 1.0 – for a fully plasticized section. The plastic coefficient is presented in the form of maps on bars.

In the non-linear analysis a load is applied in increments. All the results are accessible for the final

load increment as well as for each iteration step. Selection of the increment step is supported by the Select case component option. The current program version enables the user to choose a load step for graphical presentation of results. In tables only the results concerning the final load step are available.

Another option which allows presenting results of the non-linear analysis is Advanced analysis

diagrams (Results / Advanced / Diagrams). By these diagrams the user may present values assumed by the result quantities in the course of the iteration process. For example, these may include diagrams of force variability or of displacements of a given point or relationship between force and displacement in the course of the iterations.

Intermediate results of the non-linear analysis are saved if the Saving results after each increment

option is switched on in the parameters of non-linear analysis algorithms. For elasto-plastic analysis this option is switched on by default.

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MANUALS

A. URBAŃSKI 10.2001

II. Bar Elements in the Non-Linear Analysis in the Robot Millennium System

1. Preliminary Remarks and Assumptions

The following assumptions have been adopted while working on new bar elements: 1.1 Uniform formulation for 2D and 3D frame (2D & 3D frames, grillages). 1.2 Uniform formulation allowing for material and /or geometrical non-linearity 1.3 Standard displacement degrees of freedom at 2 extreme nodes

T

zyxzyx uuu ],,,,,[},{ φud

1.4 Optionally, the following are allowed for:

1.4.1 shear deformation included (Timoshenko’s model) 1.4.2 variable cross section, only for geometrical non-linearity 1.4.3 Winkler’s ground

1.5 There are 2 levels of geometrical non-linearity implemented which correspond to the options

existing hitherto in ROBOT, i.e.: NON second order theory, just as it is the case since RV5) PDELDA (the most accurate theory possible - large displacements and rotations,

incremental approach with geometry update, so-called Updated Lagrangian (UL) description

Thus the old PDELTA based on the Gale’s works is being substituted.

1.6 Assuming small displacements and the absence of physical non-linearity, for the limit, the results are to be identical as for standard linear elements.

1.7 The material non-linearity analysis employs the layered model and the constitutive stress-strain

principle for the uni-axial stress-strain on the point (layer) level

1.8 Shear and torsion states are treated as linearly elastic and uncoupled from axial forces and bending moments on the section level

1.9 There is a possibility to extend physical non-linearity so that it includes other options such as: 1.9.1 1D arbitrary model on the point level and a composite section 1.9.2 integral model (stresses in the elasto-plastic model determined by internal forces) 1.10 Non-linear releases and hinges may be defined only as DSC elements 1.11 All types of element loads are allowable (identically as for standard elements).

However, it is assumed that nodal forces acting on a structure are determined in the beginning of the process (i.e., changes in the transfer of element loads onto nodes resulting from geometrical or material non-linearity are ignored)

1.12 Apart from the elasto-plastic element described in this paper, it is also possible to generate elasto-plastic hinges in selected bar cross sections as an extension of the “non-linear hinges” option, described in point 5.

2. Geometry, Kinematics and Strain Approximation 2.1 Geometry, sign convention for forces, displacements, stresses and strains

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MANUALS

2.2 Basic kinematic relationships In the element local system, in the geometrically linear range, the generalized strains E on the cross section level take the following form:

(symbol x),(• indicates calculation of the differential along the direction of the bar axis)

T

zyzyox },,,,,{ Ε (1)

where: Axial strain in the bar axis:

xox u, (2)

Curvatures:

xzz

xyy

,

,

Average angles (strain) :

yxz

zxy

w

v

,

,,

Unit torsion angle: xx ,

2.3 Displacement approximation

In view of the assumption 1.4 (possibility to consider shear influence) and 1.6.1 (consistence of results obtained with the linear element), so-called physical shape functions considering shear influence have been implemented (according to WASZCZYSZYN 1990, BITNAR and others 1996, LITEWKA and others 2001). 2D bars:

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MANUALS

(3)

Shape functions and their derivatives are expressed by the formulas: (4)

]3)1(2[)21(

1])1([

)21(10

]662[)21(

1]232[

)21(

19

]3)2(2)1([)21(

1])2()1([

)21(8

]662[)21(

1)]21[(

)21(

17

]6)1(2[)21(

1]3)1(2[

)21(

16

]126[)21(

1]66[

)21(

15

]6)2(2[)21(

1]3)2(2)21[(

21

14

]126[)21(

1]66[

)21(

13

/12

/111

,

232

232

232

2

2

2

2

2

2

2

LL

LL

L

LL

L

LL

L

L

hhi xii

where:

L

x

22

6,

6

GALk

EI

GALk

EI

z

y

y

z (5)

for planes XY and XZ, respectively.

10987

10987

31

6543

6543

21

00

00

0000

00

00

0000

,)(

hhhh

hhhh

hh

hhhh

hhhh

hh

x NNuu

11 / 31

MANUALS

2.4 Kinematic relationships for the matrix notation (the geometrically linear theory) In general, when considering the influence of imposed strains

},,{T

z

T

y

T

o

o E (6)

Increment of generalized (internal) strains: (7)

GloLoc uTu , (8)

T - global local transformation matrix 2D:

2

1

106958473

6543

21

,,0,,0

,,0,,0

00,00,

u

uε

xxxx

xxxx

xx

y

z

ox

hhhhhhhh

hhhh

hh

(9)

3D:

2

1

21

106958473

106958473

6543

6543

21

00,00000,000

0,0,000,0,00

,000,0,000,0

,000,0,000,0

0,0,000,0,00

00000,00000,

u

u

E

xx

xxxx

xxxx

xxxx

xxxx

xx

z

y

z

y

ox

hh

hhhhhhhh

hhhhhhhh

hhhh

hhhh

hh

where:

T

zyxzyxzyxzyx

T

zyxzyx

uuuuuuD

uuuuD

},,,,,,,,,,,{:3

},,,,,{:2},{

222222111111

222111

21

uuu

2.5 Strains at a point (layer)

Given the generalized strains zyox ,, of the cross section, the xl strain or its increment xl at

any point of the cross section ”l” with the coordinates ll zy , , is calculated as:

lzlyoxxl yz (10)

T

ll

T

lxl yz },,1{; vEv

finally, the strain increment in the layer:

)()( oT

l

oT

lxl EuBvEEv (11)

3. Stresses and Internal Forces within an Element

o

LocL ΕuBE

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MANUALS

3.1 The constitutive principle on the point level

It is adopted in the general incremental form, where current stresses 1n

x are defined as a function of

stress for the last equilibrium n

x and the current strain increment with imposed (thermal) strains

considered

),( 11

xl

n

xl

n

xl F

(12)

based on the function )( f which describes the relationship in the process of active loading and

on the specification of the rule of unloading and reloading. In particular, it may be the elasto-plastic rule with linear hardening and the specified principle of unloading, i.e. (a) elastic, (b) plastic, (c) damage, (d) mixed. For the elastic unloading the passive and

active processes follow the same path )( f . For the remaining ones, it follows the straight line

determined by the beginning point of a given unloading process },{ UNLUNL and the unloading

module UNLD defined as

(13)

DUNLPUNLMUNLnn

n

DUNLPUNL aDDaDde

DcEDb _____ )1(:)(;:)(;:)(

.

ne is a remembered strain, for which the current active process was started, begun after exceeding 0

by stresses with the unloading ( 01 e ).

In the analysis definition of the current stiffness is necessary, which is assumed to be the following derivative:

xD

Algorithm of the constitutive rule on the point level:

Each of the cases is taken into account depending on the state obtained as a result of the previous step. Individual process phases are activated in the recurring manner:

),,,,,()(

),,,,,()(

),,,,,()(

,,:

},,{,,,:

11

11

11

111

nnnnnn

nnnnnn

nnnnnn

nnn

nnnn

DeReloadreloadstatuselseif

DeUnloadunloadstatuselseif

DeActiveactivestatusif

statusDFind

reloadunloadactivestatuseGiven

13 / 31

MANUALS

)},,,,,(

,{

}

,

),(

,

{)0)((

:

11

1

11

11

1

nnnnn

n

UNL

n

UNL

n

nnn

nnn

nn

nn

DeUnload

else

Activestatus

ed

dfD

ef

eif

Active

),,,,,(

}

),,,,,(

,

)0({

)0)((

:

11

111

1

1*1

*

nnnnn

nnnnn

UNL

nnn

UNL

nn

UNL

n

nn

DeReload

else

DeActive

De

else

DD

Dif

eif

Unload

),,,,,(

}

)},,,,,(

{

}{

)(

){0)(((

:

11

11*

*

1

nnnnn

nnn

UNLUNL

UNL

n

UNL

nn

n

UNL

nn

DeUnload

else

DeActive

else

D

if

efif

Reload

Once the equilibrium is reached, all the variables of the q model state, step n, are updated considering

the quantities obtained in the step n+1 and the initial values are assigned zeros: 01 nq

}{}{ 1 nn qq .

14 / 31

MANUALS

3.2 Calculation of forces and cross section stiffness values On the cross-section level, the vector of internal forces (stress resultants) is composed of:

T

xzyzyx

T

yzx

MQQMMND

QMND

},,,,,{:)3(

},,{:)2(

Σ

Σ (14)

Shear and torsion ST are treated as linearly elastic and not conjugated with the axial /bending forces

on the cross section level.

x

n

x

n

x

zz

n

z

n

z

yy

n

y

n

y

GIMM

GAkQQ

GAkQQ

1

1

1

(15)

Compression/tension and bending NM are generally treated as conjugate when applying the layered

approach. However, as long as the elastic state is guaranteed, i.e. until the current generalized strains fulfill the following elastic state condition (16):

1ELAZ

Z

ELAY

Y

ELAox

ox

, (16)

where:

))/((min));/((min);/(min lldll

ELAZlldll

ELAYldll

ELAox yEfzEfEf , (17)

the cross section is treated as elastic and the layered approach is not activated.

zz

n

z

n

z

yy

n

y

n

y

o

n

x

n

x

EIMM

EIMM

EANN

1

1

1

(18)

Once violation of the elastic state condition is asserted, stresses induced by axial strains and bending are calculated separately for each layer and on their basis internal quantities are calculated

Nlayer

l

lll

z

yNM

Nlayer

l

ll

n

xl

n

z

Nlayer

l

ll

n

xl

n

y

Nlayer

l

l

n

xl

n

x

A

M

MN

yAM

zAM

AN

1

1

11

1

11

1

11

v

(19)

Stiffness on the level of D cross section is calculated as follows: in the elastic state:

},,,,,{ xzyzy GIGAkGAkEIEIEAdiagD (20)

After exceeding the elastic state condition:

ST

NM

D

DD

0

0 (21)

where:

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MANUALS

Nlayer

lllll

llll

ll

ll

Nlayer

l

T

llllNM

yzyy

zyzz

yz

ADAD1 2

2

1

1

vvD (22)

},,{ xzyST GIGAkGAkdiagD (23)

3.3 Nodal force vector and element stiffness matrix They are calculated by means of the standard formulas applying Gauss quadrature, Ngauss=3.

NGAUSS

iG

iGiGiGiG

T

L

T dJWxdx10

)( ΣBΣBf (24)

NGAUSS

iG

iGiGiGiGiG

T

L

Te dJWxxdx10

)()( BDBDBBK (25)

Thus all the quantities are calculated within the sections with local coordinates as shown in the drawing:

-

1

-1 1

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MANUALS

4. Geometrical Non-Linearity

The following configurations are taken into consideration:

0B - initial configuration nB - reference configuration (the last one

for which equilibrium conditions are satisfied) 1nB - current configuration (iterated)

The basis for the element formulation is the virtual work rule which, according to BATHE 1980,

LITEWKA & WRIGGERS 2001, is saved in the following form for displacement increments:

udVeFdVCdVV

ij

n

ijV

n

ijklijklij

n

ij ,1

(26)

where: , strain increment while moving nB to

1nB , ,e constitute its parts -

correspondingly: linear and non-linear with respect to the displacement increment u , whereas is

a stress referring to the reference configuration.

4.1 The NON option It corresponds to the hitherto-applied NON formulation, i.e., second order theory The modification is only concerned with the fact, that in view of the possible material non-linearity the incremental formulation is implemented, however, without modification of element geometry. 4.1.1 Kinematic relationships Strain increments: Part:

Linear

e

Non-linear

η

(27)

Axial strain in the bar axis: xox u,

)(2/1 2

'

2

' xx wv

Curvatures: (3D)

xzz

xyy

,

,

Average angles (strain): (3D) yxz

zxy

w

v

,

,,

(3D

)

Unit torsion angle (3D) xx ,

In matrix notation

gHguΒηeE N

T

Loc 2/1 (28)

where:

T

xzxyxxxxx wvu },;,;,;,;,;,{ g gradient of displacement increment

uΓg

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MANUALS

x,NΓ a

)3(

000000000000000000000100000010000000

);2(;000010000

DDN

H

is a selection matrix. (29)

4.1.2 Nodal force vector and element stiffness matrix

L

N

T

s

L

T

L

NLn

Ln

extnnnnT

extnn

LLoc

dxN

dx

dx

0

0

int1

int1111111

)( ΓHΓK

DBBK

fffuKΣBff

KKK

(30)

4.1.3. Algorithm on the element level:

The element geometry is not modified, the local-global transformation is performed with the

use of initial transformation matrix T0

GloLoc uTu 0,

02/1 EHgguΒE T

Loc calculation of generalized strains

EΣΣΣ ,(11 nnn

, calculation of stresses (internal forces)

)( 1 nΣKK , stress stiffness matrix

NLn

Ln

extn

Locn

int1

int111 ffff

Loc

T

Glo fTf0

KKK LLoc

TKTK00

Loc

T

Glo

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MANUALS

4.2 The P-DELTA option It is a certain variant of bar description allowing for large displacements. The approach of the updated Lagrangian description is applied here. 4.2.1 Kinematic relationships Strain increments – according to CICHOŃ, 1985: Part:

Linear

e

Non-linear

η

(31)

Axial strain in the bar axis: xox u, )(2/1 2

'

2

'

2

' xxx wvu

Curvatures:

(3D)

xzz

xyy

,

,

xyxu ,,

xzxu ,,

Average angles (strain): (3D) yxz

zxy

w

v

,

,,

Unit torsion angle (3D) xx ,

4.2.2 Nodal force vector and element stiffness matrix

)3(

000000000000000000

0000

0000

000

),2(

000

0

0

)(0

1

0

int11111

DNM

NM

MMN

DNM

MN

dx

dx

dx

z

y

zy

y

y

LnT

s

L

T

L

next

nnText

nn

LLoc

O

O

ΣΣ

ΓΣΓK

DBBK

ffΣBff

KKK

(32)

4.2.3. Algorithm on the element level:

Modification of the element geometry, the local-global transformation is performed with the

use of the current transformation matrices TT1, nn

Glo

n

Loc uTu ,

0)( EuΒΒE LocNL calculation of the generalized strains according to (31)

EΣΣΣ ,(11 nnn

, calculation of stresses (internal forces) according to p.3

)( 1 nΣKK , stress stiffness matrix according to (32) for the

current stresses

int111 n

extn

Locn

fff

geometry modification

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MANUALS

Locx

nn uLL 1

,, Euler’s angles determined as for the element with the

coordinates },,,0,0,0{ wvLn , 2/)( 11 xx

);,,( TT transformation matrix for angles ,,

TTTnn 1

Loc

Tn

Glo fTf1 Transformation to the global system

KKK LLoc

TKTK11 n

Loc

Tn

Glo

After reaching the equilibrium for a given increment step, the variables stored – describing element geometry are overwritten:

LL nn 1

TT1nn

5. Elasto-Plastic Hinges Alternatively, the elasto-plastic structure work can be modeled by implementing non-linear hinges in the selected cross-sections of a bar. Properties of a hinge represented by a 2-node DSC element are defined by applying the cross section analysis algorithm described in point 3.2, assuming that the role of the generalized strains E is played by the mutual node displacements (with respect to bar local

directions) divided by the adopted (fictitious) element length L that equals the minimum cross

section height and acts as the element volume LdV . Forces and displacements of newly-

generated nodes of the DSC element constitute global degrees of freedom, in other words, they do not undergo condensation.

Algorithm on the element level. 1. calculation of generalized strains in a cross section

Glou BE

2. calculation of internal forces (stress resultants) and cross section stiffness according to point 3.2

),(

),(

1

1

EΣDD

EΣΣΣ

nn

nn

3. calculation of forces (reactions on bar ends) and DSC element stiffness

L

L

nTn

nTn

BDBK

ΣBf

11

11

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MANUALS

where:

)3(;

)2(;10000

;,

33

33 D

Dcssc

LL

x

x

T0

0TΤ

Τ

TTB

6. References 1. BITTNAR 1996, Zdenek Bittnar, Jiri Sejnota, Numerical Methods in Structural Mechanics,. ed

ASCE press, 1996 2. WASZCZYSZYN 1990, Z.Waszczyszyn, C.Cichoń, M.Radwańska, Finite element method in

structure stability, Arkady 1990 3. LITEWKA 2001: Litewka P., Wriggers P., Rakowski J., 3D beam finite element for large

displacement analysis, materials of ECCM 2001 4. BATHE 1980, Finite element nonlinear analysis 5. CICHOŃ 1985: Cichoń C., Non-linear analysis of bar structures – FEM, Monography 38,

published by Politechnika Krakowska (the Cracow Politechnics), Kraków 1988.

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MANUALS

III. Verification Manual

Elasto–Plastic Analysis in Robot Millennium

The verification examples have been tested on Robot Millennium version 15.0. The examples have been taken from: - ANSYS ® ver.5.6 - PEP Micro ® ver.4.0 - Literature

EX 1. Plastic hinge in a rectangular beam

EX 2. Limit moment analysis

EX 3. Plastic response to a suddenly applied constant force

EX 4. Plastic bending of a clamped I-beam

EX 5. Two-span frame

EX 6. Slender frame

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MANUALS

EX 1. Plastic Hinge in a Rectangular Beam

Reference: Timoshenko, Ref 4: Strength of Material, Part II, Elementary Theory and Problems, pg. 349, article 64

Source: ANSYS Verification Manual; Example VM 24

RM input file VM24.rtd

Test case:

A rectangular beam is loaded with pure bending.

Geometrical properties: b=1 in h=2 in

Material properties: E= 30e6 psi ni=0.3 Re= 36000 psi Stress-strain behavior = elastic – perfectly plastic

Loading: M = 1.0 Myp to 1.5 Myp where Myp = 24000 in-lb

Analysis Assumption and Modeling Notes

An arbitrary beam length is chosen. Due to symmetry, only half of the structure is modeled.

- 2D beam model, - bar is divided by default into six elements, - elasto-plastic element type is used, - section is divided by 5x10 layers, - load is applied by default in five increments.

Comparison of Results

According to analytical solution the beam remains elastic at M=Myp and becomes completely plastic at M=1.5 Myp. Analyzed output: stresses at the end node 2 and the solution convergence.

M / Myp Target ROBOT M Remarks

1.000 Fully Elastic Fully Elastic

1.166 Elastic-Plastic Elastic-Plastic Solution converges, Sigma_max=36000.00 psi

1.333 Elastic-Plastic Elastic-Plastic Solution converges, Sigma_max=36000.00 psi

1.500 Fully Plastic Fully Plastic Solution still converges, but it can be seen that the structure begins to collapse in the last solution step

The example confirms correctness of the solution.

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MANUALS

EX 2. Limit Moment Analysis

Reference: Crandall, An Introduction to the mechanics of solids, pg. 389, ex. 8.9



Test case:

A symmetric cross-section beam, simply supported at the beginning and totally fixed at the end, is subjected to a concentrated load P at 2/3 L point.

Geometrical properties: Total length L= 150 in Section: A=1 in2, Iy = 20 in4, h/2 =3.936 in

Material properties: E= 30e6 psi ni=0.3 Plastic properties described with Fully plastic bending moment: ML= 27777.77 in-lb Elastic moment/rotation stiffness: K= 1e12 lb/in Loading: P1 = 1000.0 lb P1 = 1388.8 lb P1 = 1390.0 lb


The beam is modeled by means of two bars. Plastic properties are modeled with non-linear hinges.

Non-linear hinge definition, rotation / moment function: Rotation (rad) Moment (lb-in)

- 2D beam model, - load is applied by default in ten increments.

0 0

1e-007 27777.77

1 27777.77

Result Comparison

Verification of the elastic and plastic moment. Examined output: - node 3 displacement UZ = maximum deflection, - reaction FZ at node 1 = R1 , - moment MY at the end of bar 2 = M2,

VALUE Target Robot MI % Robot to Target

P1 = 1000.0 lb Elastic Remarks

Deflection, in -0.02829 -0.02832 0.11

R1, in-lb 148.15 148.17 0.01

M2, in-lb -27778 -27775 0.01

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MANUALS

P1 = 1388.8 lb Solution converges

Hinge 3 Elastic Elastic MB = 27775 in-lb

Hinge 2 Plastic Plastic MC = -27777 in-lb

P1 = 1390.0 lb Solution does not converge

Hinge 3 Plastic Plastic MB = 27778 in-lb

Hinge 2 Plastic Plastic MC = -27778 in-lb

Exact results.

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MANUALS

EX 3. Plastic Response to a Suddenly Applied Constant Force

Reference: Biggs, Introduction to structural dynamics, pg. 69, article 2.7



Test case:

A mass M supported by a thin rod of area A and length L is subjected to action of a suddenly applied constant force F.

Geometrical properties: L= 144 in A=0.278 in2 M = 10 kips (m=0.0259 kips-sec2/in)

Material properties: E= 30e3 ksi Re= 162.9 ksi Stress-strain behavior = elastic – perfectly plastic

Loading: F = 1000.0 kips Force-history curve assumed as constant 1.0 in time [0.0; 0.15]


- mass of the rod is neglected, - 2D model with truss element - one calculation element, - elasto-plastic element type is used, - section is divided by 3x12 layers, - the step of time variable for which the results are stored 0.0004 sec - integration time step 0.0004/4 = 0.0001 sec

- the final time 0.15 sec is slightly greater than one cycle of vibration to be followed

Result Comparison

The diagram of node 1 vertical displacement w versus time t is displayed above.

VALUE Target Robot MI % w_max ,in 0.806 0.804 0.25

time ,sec 0.0669 0.0668 0.15

w_min ,in 0.438 0.4373 0.16

time ,sec 0.122 0.1224 0.33

Exact values.

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MANUALS

EX 4. Plastic Bending of a Clamped I-Beam

Reference: Hoff, The Analysis of Structures, pg 388, article 4.5



Test case:

A wide-flanged I-beam of length L, with clamped ends, is uniformly loaded over the whole length.

Geometrical properties: L= 144 in Section I: b=10 in h=10.6 in tf=0.9415 in

tw=0.0001 in

Material properties: E= 29e6 psi E1=5.8e6 psi Re= 38000 psi Stress-strain behavior = bilinear, elastic –plastic hardening

Loading: w1 = 2190 lb/in w1 = 3771 lb/in w1 = 9039 lb/in


The whole beam is modeled by means of one bar. An additional node is added at the center for easy tracking of deflection.

- 2D beam model, - bar is divided into 20 elements, - elasto-plastic element type is used, - section is divided into layers with web division 8 and flange thickness division 2 - load is applied by default in five increments.

Result Comparison

According to the analytical solution, at load w1 yielding has just begun at the ends, at load w2 the midpoint begins to yield and at load w3 plastic yielding has occurred. Examined output: - node 7 displacement UZ = Mid-span deflection, - moment MY at the beginning of bar 1 = End Moment, - moment MY at the middle of bar 1 = mid Moment,

VALUE Target ANSYS Robot MI % Robot

to Target w1=2190 lb/in

Mid-span deflection, in -0.16 -0.16 -0.1596 0.25

End Moment, in-lb -3.78E+06 -3.78E+06 -3.78E+06 0.00

mid Moment, in-lb 1.89E+06 1.86E+06 1.89E+06 0.00

W2=3771 lb/in

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MANUALS




W3=9039 lb/in




Exact results.

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MANUALS

EX 5. Two-Span Frame

Reference: PEP Micro, Manuel d’utilisation, pg. 7-6, ex. 7.2.1 Portique a deux travées

Source: PEP micro; Manuel d’utilisation

RM input file PEP721EP.rtd ; PEP721Hinge.rtd

Test case:

Two-span frame is subjected to plastic analysis.

Load type List Load values

uniform load 2 3 5 6 PZ=-187.00(daN/m) projected

trapezoidal load 3 PZ2=-347.00(daN/m) PZ1=0.0(daN/m) X2=1.00 X1=0.0 global projected relative

trapezoidal load 5 PZ2=0.0(daN/m) PZ1=-347.00(daN/m) X2=1.00 X1=0.0 global projected relative

uniform load 1 PX=484.00(daN/m) projected

uniform load 7 PX=82.50(daN/m) projected

nodal force 9 FX=627.00(daN) FZ=-4950.00(daN) CY=2970.00(daNm)

nodal force 10 FX=1837.00(daN) FZ=-17050.00(daN) CY=-10230.00(daNm)

self weight 1to7 PZ Negative


2D model is used.

Plastic properties are modeled with two methods: 1. Elasto-plastic elements, RM input file: PEP36EP.rtd - Material properties: Elastic – perfectly plastic

E= 210 GPa Re= 235 Mpa ro=90 kN/m3 ni=0,3 - bars are divided into elements of maximum length 1 m, - section is divided into default number of layers, - the load is applied in ten increments,

2. Non-linear hinges, RM input file: PEP36Hinge.rtd Non-linear hinge definition, stress / strain function: Strain Stress (MPa)

0 0

0,00111904 235

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MANUALS

The load is applied by default in ten increments. 0,01 235

Positions of plastic hinges:

The plastic hinges are defined at the beginning and the end of each bar at 0.01 relative length. Additionally there is a hinge defined at the point of node 10. - Material properties: Elastic

E= 210 GPa Re= 235 Mpa ro=90 kN/m3 ni=0,3 - the load is applied in ten increments.

Result Comparison

Results obtained from the applied load.

VALUE PEP micro Robot MI EP_Model

% Robot to PEP

Robot MI Hinge_Model

% Robot to PEP

displacement, cm

UX node 3 6.9591 6.4646 7.11 7.9527 14.28

UZ node 3 -5.973 -5.7564 3.63 -7.0866 18.64

UX node 6 8.7823 8.2795 5.73 10.028 14.18

UZ node 6 -7.3566 -7.4829 1.72 -8.0668 9.65

reaction , daN, cm

FX node 5 -3426.9 -3478.97 1.52 -3412.24 0.43

FZ node 5 22824 22747.37 0.34 22736.75 0.38

MY node 5 919980 961909.84 4.56 922846.6 0.31

internal forces, daN, cm

FX bar 4 end node 4 5499.2 5382.46 2.12 5371.84 2.32

FZ bar 4 end node 4 1589.9 1642 3.28 1575.24 0.92

MY bar 4 end node 4 64063 55996.64 12.59 51671.83 19.34

Overall stability (reactions) is similar. Internal forces and displacement slightly differ but there is no exact specification of material and section used in PEP model. Then the analysis is continued in order to find the structure damage load factor.

The diagram above shows that the solution with hinge model is very close to that obtained in PEP Micro. The model with elasto-plastic elements exhibits greater stiffness than the hinge model.


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MANUALS

EX 6. Slender Frame

Reference: Cz. Cichon, Non-linear structure analysis – FEM, ex. Le-Wu-Lu frame.

Source: Monography 38, published by Politechnika Krakowska (the Cracow Politechnics), Krakow 1988

RM input file R2D_LWL_frame.rtd

Test case:

Slender frame is subjected to plastic analysis taking large displacement and buckling into account.

Section ANSI W 33x130

Material properties: E= 206.8 GPa E1=0.0001 E Re= 237.7 MPa Stress-strain behavior = bilinear, elastic –plastic hardening

Load type List Load values

uniform load 12 PZ=-4.0000e+001(kN/m)

nodal force 2 13 FZ=-1.0752e+003(kN)

Additional cases analyzed, when:

• Nodal forces FX=0.01 FZ,

• Nodal forces FX=0.1 FZ,


- 2D model is used - bars are divided into 12 elements by means of node position, - section is divided into default number of layers, - load is applied in 100 increments, - solution method for basic equilibrium path – full Newton-Raphson (matrix update after each iteration) - solution method for post-buckling equilibrium path – Arc-length method

Result Comparison

Examined result: Vertical displacement of node 7 – girder mid-point.

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MANUALS

Diagrams presented in the picture: - Series 1: Target basic equilibrium path, - Series 3: Target post-buckling equilibrium path, - Series 4: Target post-buckling equilibrium path when FX=0.01FZ, - Series 5: Target post-buckling equilibrium path when FX=0.1 FZ , - Fx=0 / Full-NR: Basic equilibrium path, Robot solution obtained with full Newton-Raphson method, force

increment control, - Fx=0 / AL: Post-buckling equilibrium path, Robot solution obtained with Arc-length method, node 7 UZ

displacement increment control, - Fx=0.01Fz / AL: FX=0.01 FZ, post-buckling equilibrium path, Robot solution obtained with Arc-length

method, node 13 UX displacement increment control, - Fx=0.1Fz / AL: FX=0.1 FZ, post-buckling equilibrium path, Robot solution obtained with Arc-length method,

node 7 UZ displacement increment control.


I. OPERATION OF THE OPTION 2 fileII. BAR ELEMENTS IN THE NON-LINEAR ANALYSIS IN THE ROBOT MILLENNIUM...

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