Page 1

7th International Conference on Structures in Fire

M. Fontana, A. Frangi, M. Knobloch (Eds.)

Zurich, Switzerland, June 6-8, 2012

A CO?TI?UUM-MECHA?ICAL MODEL FOR FE-SIMULATIO?S OF

STEEL CO?STRUCTIO?S I? FIRE

Sebastian Hauswaldt*, Manfred Korzen* and Stefan Hartmann**

* BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin,

Germany. e-mails: sebastian.hauswaldt@bam.de, manfred.korzen@bam.de

** Institute of Applied Mechanics, Clausthal University of Technology, Adolph-Roemer-Str. 2a,

38678 Clausthal-Zellerfeld, Germany. e-mail: stefan.hartmann@tu-clausthal.de

Keywords: Constitutive Equation, Viscoplasticity, Endochronic Plasticity, UMAT, Finite Element

Method, Fire Resistance

Abstract. This paper presents a thermoviscoplastic constitutive model within the framework of

continuum mechanics for the description of the behavior of structural steel subjected to fire loading. The

model is represented through a set of ordinary differential equations and belongs to the class of so-called

unified theories which do not separate plastic and creep strains. The material parameters are identified

for the structural steel S 235 on the basis of tension and compression tests at constant temperature with

varying strain rates and verified against transient creep test with varying temperature. For effective

numerical simulations with the finite element program ABAQUS the model is prepared with a stress

algorithm and consistent tangent operator. The simulations are compared with experimental data of fire

resistance tests on columns.

1 I?TRODUCTIO?

For the design of steel structures in fire the common material model of EC3-1-2 [1] is formulated as

an one-dimensional rate-independent relationship between stress and mechanical strain. It can be shown

that the material behavior of carbon steel is well described for monotonically increasing temperatures and

constant loading.

But in case of fire, structures are also exposed to varying thermal and mechanical boundary

conditions requiring additional considerations. For the simulation of these processes a model should be

chosen that exhibits inelastic properties and is capable to describe creep and relaxation at any heating or

cooling condition.

Therefore, a thermoviscoplastic material model for the simulation of structural steel members under

fire conditions has been developed.

2 A THERMOVISCOPLASTIC MATERIAL-MODEL

Within the framework of small deformations the deviatoric part TD of the stress tensor T is assumed

to be additively decomposed into two parts, i.e.

with an equilibrium stressTeq and an overstressTov with the deviatoric decomposition

1 3 ( )

=−

TT T Itr

with the trace

( ).

=

T

ii

trT

The spherical part of the stress tensor is assumed to be defined using the trace operator

classical thermoelastic relationship

=+

TTT

DD

eq

D

ov

(1)

D

(.)tr

via the

Page 2

Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann

( )T3()

m

tr Ktr

=

E

(2)

with the mechanical strain tensor

the absolute temperatureθ and reference temperature 0,

of the bulk modulus K and thermal expansion coefficient .

In view of assigning specific material properties to the stress components in eq. (1), see [3], Teq will

be assumed to be a rate-independent functional whereas Tov has to be a rate-dependent functional of the

process history.

Based on the theory of endochronic plasticity [4] the equilibrium stress is assumed to be described by

[]

m0

α θθ

=−−

EE1

(3)

θ the strain tensor E and the material parameters

α

where ()ɺdenotes the material time derivative, with the strain deviator

the limit shear modulus Gt and the hardening parameter

differential equation

2[2] (0)0

β

=−−=

T

ɺ

E

ɺ

TET

ɺ

z

D

eq

DD

eq

DD

eqst

GG with

(4)

,ED the initial shear modulus Gs,

β The variable z is defined through the

.

with the additional material parameter κ constrained to0

rate-independent functional for the equilibrium stress. It is characterized by the fact that the strain is not

decomposed into elastic and plastic parts and that no yield surface and corresponding loading conditions

have to be defined as it is the case in classical elasto-plastic constitutive models with a yield function.

The overstress is assumed to be defined by the differential equation

2

κ

Gs

=−

E

ɺ

T

ɺ

ɺ

z

DD

eq

(5)

1

κ≤< (see Fig. 1). Eqs. (4) and (5) define a

with the instantaneous shear modulus G0 and relaxation timeλ , which is given as a function of the

overstress

D

ov

D

ov

DD

eq

D

ov0

1

λ

2G with (0)0

= −+−=

T

ɺ

TE

ɺ

T

ɺ

T

(6)

D

ov2

D

ov01

3

a

)aa 1 tanh

−

a

λ

+

=+

T

(T

. (7)

Taking into account that TɺD

of viscoelasticity with respect to the equilibrium stress, i.e. TD

or for a relaxation process at constant strain as time is going to infinity.

All material parameters, i.e. K,

temperature . θ

eqis defined by (4), eq.(6) can be interpreted as a modified Maxwell model

ovvanishes asymptotically for slow motions

,

α Gs, Gt,

,

β

, κ G0, a0, a1, a2 and a3 may depend on the

3 U?IAXIAL STRESS CASE

For analyzing the proposed constitutive model in more detail, we apply it to the situation of simple

extension with a uniaxial stress state.

Taking into account the relationships

Page 3

we arrive at we arrive at we arrive at we arrive at we arrive at we arrive at we arrive at

w

response of the equilibrium stress

with the

Fig. Fig.Fig.

a)a)

elasticity E

for different input functions characterized by the fact for different input functions characterized by the fact

1 ε ɺ

ε ɺ

0,

=

0,

creep for creep for creep for creep for creep for σ

σ ɺ

σ =

0

=

and nd nd an

with the mechanical s

response of the equilibrium stress

with thewith thewith thewith the associatedassociatedassociatedassociatedassociatedassociated material parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Ematerial parameters Es, E

Figure 1. Figure 1. Figure 1. Figure 1. Figure 1. Figure 1. Stress

elasticity Eelasticity E

for different input functions characterized by the fact

1 discontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude odiscontinuously the magnitude or type of input rate

ith the mechanical s

response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress response of the equilibrium stress

A springA spring

2a. 2a.

Please note Please note

elasticity E

for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact for different input functions characterized by the fact that they change after a period of constant strain rate

type of input rate

rate from rate from rate from rate from ε ɺ

ith the mechanical sith the mechanical sith the mechanical sith the mechanical sith the mechanical sith the mechanical sith the mechanical sith the mechanical sith the mechanical strain

A springA springA springA spring-dashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole onedashpot representation of the whole one-dimensional model

Please note

elasticity Eelasticity Eelasticity E0.

Please note Please note Please note Please note

StressStress-

. The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of The material response of the total stress

Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Figure 2. a) Spring

σσ

ɺeq

σσσσσσσσσσσσσσσσσσσσσσσσσσσσσεεεεεεεεεεεεεεεεεεεεεεεε

ɺ

εεεεεεεσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσ

σσ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

σσ

σ

σσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

εε

ε

εεεε

ε

ε

0

ε

0

ε

ɺ

εε

ε

εε

ε

εε

ε

εε

ε

εεεε

ε

εε

ε

σσ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σ

ɺ

σσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

σσ

σ

σσ

σ

σσ

σ

σσ

σ

σσσσ

σ

σσ

σ

-strain curves strain curves strain curves strain curves strain curves strain curves of the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading aof the equilibrium stress (see eq. (9)) for monotonic loading at constant temperature

that for fast lthat for fast lthat for fast lthat for fast lthat for fast lthat for fast lthat for fast lthat for fast loadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of

the total stress the total stress the total stress the total stress the total stress the total stress the total stress

that they change after a period of constant strain rate

type of input ratetype of input rate. Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to

ε ɺ to to

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq

ov

ov

ov

ovov

ov

ovov

ov

ovov

ov

ovovovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

ov

ov

ov

ovov

ov

ovov

ov

ovov

ov

ovovovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

m

m

m

mm

m

mm

m

mm

m

mmmm

m

mm

m

mm

m

mm

m

mm

m

mm

eq

eq

eq

eqeq

eq

eqeq

eq

eqeq

eq

eqeqeqeq

eq

eqeq

eq

eqeq

eq

eqeq

eq

eqeq

eq

eqeq

ov

ov

ov

ovov

ov

ovov

ov

ovov

ov

ovovovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

ov

ovov

eqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq

==============================================================

= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −= −+++++++++++++++++++++++++++++++−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−===============================

train train

SSebastianebastianebastianebastian

ββββββββββββββββββεεεεε

ɺ

εεεεεεεεεεεεεσσσσσσσσσσσσσ

εεεεε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

E

ε

t m

E

ε

t mt m

λλλ

−−−−−−−−−−−−−−−+++++++++++++++−−−−−−−−−−−−−−−

1

εε

ε

εε

ε

εε

ε

εε

ε

εε

ε

εε

ε

α θα θ

α θ

α θα θ

σ

σeq

θθ

θ

θθ

θ

θθ

i.e. eq.i.e. eq.

, Et,

SpringSpringSpring-

1

EEEEEEEEEEEEEEEEEEEEEEEE

1

GGGGGGGGGGGGGGGGGGGGGGGG

1

KKKKKKKKKKK9K9K9K9K9K9K9K9K9K9K9K9K9K

1

EEEEEEEEEEEEEEEEEEEEEEEE

1

GGGGGGGGGGGGGGGGGGGGGGGG

1

KK9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K9K

sssssssssssssssssssssssssssssssssssssssssstttttttttttttttttttttttttttttttttttttttttt

sgn(sgn(sgn(sgn(sgn()))))

EEEEEEEEEEEEEEEEEEEEEEEE

========−−−−−−−−−−−−−−−−

i.e. eq.i.e. eq.(9

m

an increaincreaincreaincrease of strainse of strainse of strainse of strainse of strainse of strainse of strain-rate from

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

=====================+++++++++++++++++++++=====================+++++++++++++++++++++

ssssssssssssseqeqeqeqeqeqeqeqeqeqeqeqeq t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t m

ε

t mt m

33333333333333333333339999999993333333333333333333333

β

β

ββ

β

ββββ

β

βε

ε

εε

ε

ɺ

εεεε

ε

εσ

σ

σσ

σ

σ

κ

GG

mmmmmmmmmmmmmmmeqeqeqeqeqeqeqeqeqeqeqeqeqeqeq

EEEEEEEEEEEEEEEEEEEEEwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwithwith

-dashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representationdashpot representation

Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred

−−−−−−−−−−

sgn( )

9

K9

K9

K9

K

K

933333

[

α θ

α θ

eq,

sgn( )

1

−

GGG

type of input ratetype of input ratetype of input ratetype of input ratetype of input ratetype of input rate

sgn( )

−

α θ

α θ

sgn( )

−−−−

]

]and the total strainand the total strainand the total strain

9), for monotonic loading is depicted in F

andand ββ

to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.to identify their physical significance.

0

θ

oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of

the total stress σ

that they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain rate

. Particular focus has to be drawn to . Particular focus has to be drawn to

1 ε ɺ

σ

κ

, κκ and

σ

s

anfredanfred

σ

σ

σ

rate from

andandand

σ

σ

σ

σ

σ

and the total strain

), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in F), for monotonic loading is depicted in Fig.

Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan

and the total strainand the total strainand the total strainand the total strainand the total strainand the total strainε

b) b) Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.Material response for different inputs.

dimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional model (Eqs.

b)

oadings the model responds always with the instantaneous modulus of

σ at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.

that they change after a period of constant strain ratethat they change after a period of constant strain rate

. Particular focus has to be drawn to

1 ε .

oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of

at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.

that they change after a period of constant strain rate

. Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to . Particular focus has to be drawn to relaxation

ε in in in σ

tefantefan HartmannHartmannHartmannHartmannHartmann

oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of

at constant temperature is depicted in Fig.

that they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain ratethat they change after a period of constant strain rate

withwithwith

σ -

oadings the model responds always with the instantaneous modulus of

at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig.at constant temperature is depicted in Fig. 2

-direction (direction (direction (direction (direction (direction (cf.

oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of oadings the model responds always with the instantaneous modulus of

(Eqs. (Eqs. (Eqs. (9

(0)(0)(0)(0)0000

(0)(0)(0)(0)0000

(9)

t constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperaturet constant temperature.

)-(101010))

cf. [5]). [5]). [5]). [5]). The material The material

ig. 1 together

is presented in is presented in is presented in is presented in is presented in is presented in is presented in is presented in

The material

1 together 1 together 1 together 1 together 1 together

relaxationrelaxationrelaxationrelaxationrelaxation

The material The material The material The material

(10)(10)(10)

(8)(8)

(9)(9)

2b

Page 4

44

20

S

Composition analysed with aComposition analysed with aComposition analysed with a Composition analysed with a Composition analysed with a Composition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with aComposition analysed with a Composition analysed with aComposition analysed with a Composition analysed with aComposition analysed with a

identified

values at the

equilibrium stress.equilibrium stress.equilibrium stress. equilibrium stress. equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.equilibrium stress.

The instantaneous

aforementioned tests

assumption of assumption ofassumption of assumption of assumption ofassumption ofassumption ofassumption of th

20

S 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens 235 specimens were

identified

values at thevalues at the values at thevalues at thevalues at thevalues at thevalues at the end of hold timesend of hold timesend of hold timesend of hold timesend of hold timesend of hold times end of hold timesend of hold timesend of hold times end of hold times

The instantaneous

aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests aforementioned tests (see

EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE? EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?EXPERIME?TS A?D IDE?TIFICATIO? OF THE MA

The The

, 300, 4, 300, 4, 300, 400, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 00, 500 and 20, 300, 4

In Fig.In Fig.In Fig.In Fig.In Fig. 3

identifiedidentifiedidentifiedidentified equil

The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous The instantaneous

The material parameters asmaterial parameters asmaterial parameters asmaterial parameters asmaterial parameters as material parameters asmaterial parameters asmaterial parameters as material parameters asmaterial parameters as material parameters as material parameters as

equilequilequilibrium stressibrium stress ibrium stress ibrium stress ibrium stressibrium stress ibrium stress

Table 2. Measu Table 2. MeasuTable 2. MeasuTable 2. Measu Table 2. MeasuTable 2. MeasuTable 2. MeasuTable 2. Measured instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous red instantaneous Young’s

0.150.15 0.15

3 you can see the stress you can see the stressyou can see the stress you can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stressyou can see the stress you can see the stressyou can see the stressyou can see the stress-strain diagram with the strain rates for

based on eq. (

tend toward equilibriumtend toward equilibrium tend toward equilibriumtend toward equilibriumtend toward equilibrium tend toward equilibriumtend toward equilibriumtend toward equilibriumtend toward equilibrium tend toward equilibriumtend toward equilibrium tend toward equilibrium

Figure 3. StressFigure 3. Stress Figure 3. StressFigure 3. Stress Figure 3. StressFigure 3. Stress Figure 3. StressFigure 3. Stress-strain

CC

the temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of e temperature dependence of Poisson’s ratio

werewere

Si

0.20.2

Young’s modulus Young’s modulus

(see (see Tab.Tab.Tab. 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus 2). Bulk and shear modulus are

TemperatureTemperatureTemperature Temperature

20

300

400

500

600 600

600

performedperformedperformed performedperformed performed at BAM

spark emissispark emissispark emissispark emissispark emission spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is noton spectrometer is not on spectrometer is noton spectrometer is noton spectrometer is noted.

Si

SSebastianebastianebastianebastian

600

Young’s modulusYoung’s modulusYoung’s modulusYoung’s modulusYoung’s modulus Young’s modulusYoung’s modulus Young’s modulus

300

400

500

600

°C with°C with°C with °C with°C with varying

at BAM at BAMat BAMat BAM on a servo

Mn

0.490.490.49

spark emissispark emissi

MnMn

based on eq. (based on eq. (based on eq. (

tend toward equilibrium

traintrain-

Temperature

20

300

400

500

functions offunctions offunctions offunctions of

varyingvarying varyingvarying strain rate

Temperature Temperature

Table 4. Chemical Composition STable 4. Chemical Composition S

P

0.0190.0190.019 0.0160.0160.0160.016 0.019

-diagramiagramiagramiagram

Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, M Hauswaldt, MHauswaldt, MHauswaldt, Manfred

Table 4. Chemical Composition S

P SS

strain diagram with the strain rates for

based on eq. (based on eq. ( based on eq. (based on eq. (99) for for for the tested temperature the tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperature the tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperaturethe tested temperature.

TIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MA TIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MA TIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MA TIFICATIO? OF THE MATIFICATIO? OF THE MATIFICATIO? OF THE MATERIAL

functions offunctions of functions of the

Table 4. Chemical Composition STable 4. Chemical Composition S Table 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition S Table 4. Chemical Composition S Table 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition S Table 4. Chemical Composition STable 4. Chemical Composition STable 4. Chemical Composition S

strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for strain diagram with the strain rates for a

νννννννννννννθθθθθθθθθθθθθ

213000

204500

194500

177000

152500152500152500 152500152500

of strain of strainof strainof strain-

E0

= 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054 = 0.292+0.000054

EE0

213000

204500

194500

177000177000177000177000

on a servoon a servo on a servoon a servoon a servo-

0

Young’s Young’s Young’s Young’s modulus and modulus and modulus and modulus and modulus and modulus and modulus and calculated

213000

204500

194500194500194500

the

strain rate strain ratestrain ratestrain ratestrain rates

as a function ofas a function of as a function ofas a function ofas a function ofas a function ofas a function ofas a function ofas a function of as a function of

213000

204500204500

the temperature temperature temperature temperature temperature temperature temperature were identified on the basis of

and hold times

hydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machine

ed.

oisson’s ratiooisson’s ratio oisson’s ratio oisson’s ratiooisson’s ratiooisson’s ratiooisson’s ratio according to

213000

anfredanfred

0.050.050.05

-controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600controlled tensile test with hold times at 600 controlled tensile test with hold times at 600controlled tensile test with hold times at 600 °C.

Cr Cr

Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan

-hydraulic testing machinehydraulic testing machinehydraulic testing machine hydraulic testing machinehydraulic testing machinehydraulic testing machinehydraulic testing machine hydraulic testing machinehydraulic testing machinehydraulic testing machine. In Table 1 the Chemical

s andand

171502

177680

173688

163012

144678 144678144678144678

Mo

0.010.010.01

KK

171502

177680

173688

163012163012163012

MoMo

states statesstates

171502

177680

173688173688

were identified on the basis of

hold times hold timeshold timeshold times hold times of 9000 of 9000of 9000of 9000of 9000 seconds

according to according to according to according to according to according to according to [7

171502

177680

were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of were identified on the basis of tensile

235235

0.07 0.070.07

are

calculatedcalculatedcalculatedcalculated bulk and shear modulus. bulk and shear modulus.bulk and shear modulus. bulk and shear modulus.bulk and shear modulus. bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.bulk and shear modulus.

NiNi

whichwhich whichwhich are

temperature

calculated calculated calculated calculated calculated calculated from these values

7], i.e. , i.e., i.e.

tefantefan Hartmann HartmannHartmann HartmannHartmann

in %.in %. in %.

temperature temperaturetemperature temperaturetemperature temperature is

<0.01<0.01 <0.01<0.01

TERIAL TERIALTERIALTERIALTERIALTERIAL PARAMETERS PARAMETERSPARAMETERS PARAMETERSPARAMETERS PARAMETERSPARAMETERS PARAMETERSPARAMETERSPARAMETERS

AlAl

82332

78151

73939

67108

57548575485754857548

. It was assumed that

are are interpreted as points of theinterpreted as points of the interpreted as points of theinterpreted as points of theinterpreted as points of the interpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of theinterpreted as points of the interpreted as points of the interpreted as points of theinterpreted as points of the interpreted as points of theinterpreted as points of the

GG0

82332

78151

73939

6710867108 67108

It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that It was assumed that

82332

78151

7393973939

0.05 0.050.050.05

a 600600600 °C tensile test °C tensile test°C tensile test °C tensile test°C tensile test°C tensile test°C tensile test °C tensile test and t

from these values from these values from these values from these values from these values from these values from these values from these values from these values from these values with the

82332

78151

CuCu

is

seconds

In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical In Table 1 the Chemical

determined determined determined determined determined determined determined from the

secondssecondsseconds. T

0.0030.003 0.003

NN

The tests he tests he tests he tests he tests of the

°C. °C.

tensiletensile tensile

the stressthe stressthe stressthe stressthe stress

from the from the

with the with the with the with the

tests tests

of the of the of the

and tand t and the

from the from the from the

(11)(11)(11)

tests at at

he

Page 5

corresponding

assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent

The overstress has to be identified on the basis of the difference between equilibrium stress and measured

total stress according

parameterparameterparameterparameter parameterparameterassas0.01

=

as 0.01as0.01as 0.01as is is cconstant andonstant andonstant andonstant and onstant andonstant andonstant and temperature temperaturetemperaturetemperaturetemperaturetemperaturetemperature-independent.independent. independent.independent.independent.independent. independent.

corresponding corresponding corresponding

assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent assumed to be constant and temperature independent assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent assumed to be constant and temperature independent assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent assumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independentassumed to be constant and temperature independent.

The overstress has to be identified on the basis of the difference between equilibrium stress and measured

total stress accordingtotal stress accordingtotal stress according total stress accordingtotal stress according total stress according

0 a

The The The material parameters of the

corresponding corresponding corresponding corresponding parameter parameter parameter parameter parameter parameter functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig.functions of temperature are depicted in Fig. 4. The parameter

The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured

total stress according

0=

material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the material parameters of the

Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature Material parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperatureMaterial parameters of the equilibrium stress as a function of temperature.

The overstress has to be identified on the basis of the difference between equilibrium stress and measured

total stress accordingtotal stress accordingtotal stress accordingtotal stress according toto eqeq. . (10).(10). (10). The materialThe materialThe materialThe materialThe materialThe materialThe material- -parameters aparameters aparameters aparameters a parameters a parameters aparameters a1

The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured The overstress has to be identified on the basis of the difference between equilibrium stress and measured

Figure Figure Figure Figure 5.

SSebastian ebastianebastian ebastian

. Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the Material parameters of the over

Hauswaldt, M Hauswaldt, MHauswaldt, MHauswaldt, M Hauswaldt, MHauswaldt, MHauswaldt, Manfred

equilibriumequilibrium equilibriumequilibriumequilibriumequilibrium stress are stress are stress are stress are stress are identified identified identified identified identified identified using

anfredanfred Korzen and SKorzen and SKorzen and SKorzen and SKorzen and S Korzen and SKorzen and Stefan

over overstress as a function of temperstress as a function of temperstress as a function of temper stress as a function of temperstress as a function of temper stress as a function of temperstress as a function of temperstress as a function of temper stress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperstress as a function of temperature

1, a2

tefantefan HartmannHartmannHartmann HartmannHartmann

using using using

2 and aand a and aand a3

eq.eq. (9). The paramet

4. The parameter4. The parameter4. The parameter4. The parameter 4. The parameter4. The parameter 4. The parameter4. The parameter 4. The parameter

are depicted in Fig. 5 are depicted in Fig. 5are depicted in Fig. 5 are depicted in Fig. 5are depicted in Fig. 5are depicted in Fig. 5 are depicted in Fig. 5 are depicted in Fig. 5are depicted in Fig. 5 are depicted in Fig. 5are depicted in Fig. 5 are depicted in Fig. 5.

(9). The paramet (9). The paramet(9). The paramet (9). The paramet (9). The paramet(9). The paramet(9). The paramet (9). The parameters

atureature.

.

ers

κκ =

0.7 0.7 0.7 0.7 0.7 is

ers withwith with

. The The The

the the

is

Page 6

Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann

5 IMPLEME?TATIO? OF THE CO?STITUTIVE MODEL I?TO FE-SOFTWARE

The stress state at time tn+1 develops from

with the time increment t △ and the stress rate

The spherical part of the stress tensor is

.

1

nn

t

+ =+

TTT

△

(12)

.

∆=

TT △t

1,1

()3()

n m n

trKtr

++

=

TE

.

(13)

According to eq. (4)

,1,

+

=−∆TTT

D

eq

D

eq n

D

eq ncan be determined using a Backward Euler method leading

to

?

L

,1

1

β

22 0 :

=

DD

eq n

T

D

eq

D

n

D

eqst

GzG

+

∆−+ ∆−− ∆=

ETET

△

(14)

Here

1

+

∆=−

EEE

DD

n

D

nis prescribed. From eq. (5) follows the incremental formulation of the arc-length

2

2

s

2

3

4

κ

G

κ

G

=∆∆−∆∆+∆⋅∆

⋅

⋅

EEETTT

△

DDDD

eq

D

eq

D

eq

s

z

. (15)

According to eq.(6), the overstress

,1

D

ov n+

T

can be calculated with

,10,1

D

ov n

T

D

n

DD

eq n

T

G

t

λ

+

λ

++

=+∆−

TE

△

.

(16)

For effective numerical simulations with the finite element method in addition to the stress

formulation we need the consistent tangent operator. The tangent operator is needed to develop the

structure stiffness matrix in a finite element analysis.

Since the stress computation is based on an iterative procedure, the implicit function theorem has to

be applied requiring the application of the chain-rule applied to eq. (14), see [6] and the literature cited

therein.

?

L

E

?

L

∂

∂

∂

T

+=

∂

T

d

0

E

D

eq

D

eq

d

. (17)

?

L E can be found using Gâteaux-derivatives. With the derivative of the arc-length with respect to the

strain tensor

∂∂

1

1 2

2

△

z

( )[ ]H2[ ]H

3

κ

G

+

=∆−∆

EET

△

DD

eqn

s

D z

(18)

we obtain

1

,1

1

3

ˆ(

)[ ]

H

22 [ ]

1 1

3

β

22[ ]H

β

κ

G

+

+

=+−⊗

−+ ∆−⊗∆−∆

EII H

TTEET

△

△

nst

D

eq n

D

eq

D

n

DD

eqt

s

z

DLGG

G

z

I

.

(19)

Page 7

Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore Furthermore the derivat

is required and the expressionis required and the expression is required and the expression is required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expressionis required and the expression∂

After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. After rearranging eq. (17)

numerical solver with

rightrightright

The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of The tangent operator of the

is is derived from eq.derived from eq.

The model

FORTRAN

tangent operatortangent operatortangent operatortangent operatortangent operatortangent operatortangent operator

FORTRAN

tangent operatortangent operator

66

strain

FFig. 6big. 6big. 6big. 6b.

a)a)

of of of -100100100 MPa

numerical solver with numerical solver with numerical solver with

right-hand sides.hand sides.hand sides.hand sides.hand sides.hand sides.

FORTRAN FORTRAN FORTRAN FORTRAN FORTRAN

strainstrainstrain-temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests temperatures curves correspond to the observations of transient creep tests of S

Figure 6. Figure 6. Figure 6. Figure 6. Figure 6. a)

tthe tangent operator ofhe tangent operator of

numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with numerical solver with

TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP TRA?SIE?T CREEP

In Fig.In Fig.In Fig.In Fig.In Fig. 6

Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests Within the framework of compression and tensile tests

MPa MPa MPa have been have been have been have been have been have been carried out. carried out. carried out. carried out. carried out. carried out. carried out. The temperature rate was 7.2

DDDDDDDDDDDDDDDDDDDDDDDDGGGGGGGGGGGGGGGGGGGGGGGG

derived from eq.

The model The model The model The model The model The model is adapted to

subroutinesubroutinesubroutinesubroutinesubroutines

?LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEETTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEHHHHHHHHHHHHHHHHHHHHH[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H [ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ]H[ ] .

derived from eq.derived from eq.derived from eq.derived from eq.derived from eq.derived from eq.derived from eq. (1

?()[ ])[ ])[ ])[ ])[ ])[ ] =)[ ] =

he tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator ofhe tangent operator of

((((((

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460 at 170460 at 170460 at 170460 at 170460 at 170 MPa [8].

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1

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235 235

K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of K/min till the maximum temperature of

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235 transientransientransientransientransient creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress creep tests at constant stress

[2] using the UMAT interface.

algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated algorithm and the associated

using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface. using the UMAT interface.

are presentedpresented

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460460 [8]

II

. The simulated The simulated

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(20)(20)(20)

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(24)(24)(24)

a

Page 8

Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann

600 °C was reached after 5000 seconds. The following cooling phase with a temperature rate of -

7.2 K/min. ended at 120 °C when natural cooling was not fast enough.

Temperature, uniaxial stress and strain were recorded and compared to results of a simulation of the

test as depicted in Fig. 7. In the heating phase the calculated strain increases to higher values then in the

transient creep test. At higher temperatures when cooling down the simulated strains are in good

agreement with the experiment.

Figure 7. Transient creep test and simulation of S 235.

7 COLUM? TESTS

Furthermore, component simulations are compared with the results of two column fire tests. The

steel columns with a height of 3.56 m were fixed at the top while the bottom end was free to move

vertically. The mechanical load applied at the center of the bottom of the column was constant. By the use

of a servohydraulic testing machine an elastic foundation was applied additionally [10]. Axial loading and

stiffness of the foundation are presented in Tab. 3.

Table 3. Test Program Columns

Specimen Load

HEA 140 492 kN

HEB 180 1052 kN

Stiffness

47 kN/mm

69.5 kN/mm

Page 9

EN

1111 11 measuring

The temperature fi

using the measured surface temperatures at

thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according to

using the measured surface temperatures at

thermal conductivity, density and heat capacity according thermal conductivity, density and heat capacity according

Figure 8.

simulation csimulation csimulation csimulation c simulation csimulation csimulation c) T

thermoviscoplastic material

application point at the bottom of simulation and experiment are compared in Fig.

time and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are intime and temperature of maximum displacement and zero passage are in good agreement.

EN EN 1363 1363

measuring

The temperature fi

using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at using the measured surface temperatures at Dirichlet boundary conditionsusing the measured surface temperatures at

Figure 8. Figure 8. Figure 8. Figure 8. HEA

a)a)

a)a)

thermoviscoplastic material

application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig. application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig.application point at the bottom of simulation and experiment are compared in Fig. 10. It can be seen that

The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to

1363

measuring

The temperature fi The temperature fiThe temperature fiThe temperature fiThe temperature fiThe temperature fiThe temperature fiThe temperature field of the eld of the eld of the eld of the eld of the eld of the

With the use of the calculate temperature field the column tests were simulated with the presented

thermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic material thermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic materialthermoviscoplastic material model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load model. For the evaluation of the results the vertical displacements of the load

With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented With the use of the calculate temperature field the column tests were simulated with the presented

a)

The furnace was heated up according to

1363-1

measuring measuringmeasuring stationsstationsstationsstationsstations ((see Fig.see Fig.see Fig.see Fig.

HEAHEAHEA 140

Temperature field emperature field emperature field emperature field emperature field emperature field emperature field emperature field at deformed column

The furnace was heated up according to

1 standardstandardstandardstandardstandard [9]. [9]. [9]. The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30

8a8a).

columnscolumnscolumnscolumnscolumns has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element

Dirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditionsDirichlet boundary conditions a

The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to The furnace was heated up according to the so called uniform

The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30 The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30

140140: a)a) TemperatureTemperatureTemperatureTemperatureTemperatureTemperatureTemperature m

SSebastianebastianebastianebastian

Figure Figure Figure Figure

b)

Figure Figure Figure Figure 9

at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at deformed column at

b)

The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30

.

has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element has been calculated via the finite element method

measuring easuring easuring easuring easuring stations b)

1010. Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement . Vertical displacement a)

Hauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, MHauswaldt, Manfred

9. Steel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEASteel Temperatures a) HEA

The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30The measured surface temperatures were recorded with 30

tations b)tations b)tations b)

at 1600

the so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniformthe so called uniform-temperature

anfredanfred

tations b)

160016001600 sec.

b)

b)

c) c)

V. V. M

Korzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and SKorzen and Stefan

b)

Mises s

sec. sec. of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation of the simulation d)

b)

ises sises stress

a) HEAHEAHEA

to [1].[1].[1].

tresstresses ofes ofes of

140 b) HEB140 b) HEB140 b) HEB140 b) HEB140 b) HEB140 b) HEB 180

140140 b)

tefantefan HartmannHartmannHartmannHartmannHartmann

b) b) HEBHEBHEB 180

temperaturetemperaturetemperaturetemperaturetemperaturetemperature-time

deformed columndeformed columndeformed column

d) Deformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed columnDeformed column

d) d)

180180.

good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.good agreement.

and the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of nd the material properties of

deformed columndeformed columndeformed columndeformed columndeformed column

180180.

timetime-

10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that 10. It can be seen that

-curvecurve

thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples thermocouples at

methodmethodmethodmethod (

at 255at 255at 255 sec.

curvecurve according to according to according to according to according to according to according to

sec.

after test.after test.after test.after test.after test.

(ABAQUSABAQUSABAQUSABAQUSABAQUSABAQUS)

sec. of the of the of the of the

at

Page 10

Sebastian Hauswaldt, Manfred Korzen and Stefan Hartmann

8 SUMMARY A?D CO?CLUSIO?S

A so-called unified thermoviscoplastic constitutive model for the simulation of the behavior of

structural steel is presented. Due to the modular concept with an additive decomposition of the stress into

an equilibrium stress and an overstress specific material properties of rate-independent classical plasticity

and rate dependent viscoelasticity can be associated to experimental observations.

Based on this concept the identification of the temperature-dependent material parameters for

structural steel S235 are realized within a series of strain-controlled tensile and compression tests with

varying strain-rates at constant temperature between 20 °C and 600 °C.

The modular concept of the constitutive model influences also the numerical solution strategy.

Whereas the numerical integration of the overstress might be realized by an explicit algorithm the

corresponding procedure for the equilibrium stress are treated via an implicit Backward-Euler algorithm.

This stabilizes essentially the computational procedure. The necessary consistent tangent operator splits

into two parts as well. Whereas the tangent operator for the overstress is written analytically via Gâteaux

derivative, the corresponding operator of the equilibrium stress is solved numerically. The numerical

implementation has been programmed as a user subroutine UMAT for the FE-program ABAQUS.

Comparing simulation results with experimental data it is shown that stress-controlled transient creep

tests are described with sufficient accuracy. The comparison of experimental data of fire resistance tests

on columns with numerical simulations show that the presented constitutive model is applicable without

problems and is capable to describe the column behavior quite reasonably. In contrast to rate-independent

models as in [1] the presented model is able to describe properly the material behavior as a function of the

heating-rate.

REFERE?CES

[1]

EN 1993, Eurocode 3: Design of steel structures - Part 1-2: General rules – Structural fire

design, CEN, Brussels, Belgium, December 2010.

ABAQUS 6.11-1: Abaqus Analysis User’s Manula Version 6.11-1, Dassault Systèmes Simulia

Corp., Providence, RI, USA, 2011.

Haupt, P.: “On the Mathematical Modelling of Material Behaviour in Continuum Mechanics,

Acta Mechanica”, 100, 129 -154, 1993.

Valanis, K.C.: “Fundamental Consequences of a New Intrinsic Time Measure Plasticity as a

Limit of the Endochronic Theory”, Archives of Mechanics 32, 171-191, 1980.

Korzen, M.: “Constitutive Equations for Structural Steel Subjected to Fire”, Applications of

Structural Fire Engineering – Proceedings of International Conference, F. Wald, K. Horová, J.

Jirku (eds.), Prague, 2011.

Hartmann, S.: "A remark on the application of the Newton-Raphson method in non-linear finite

element analysis", Computational Mechanics, 36 (2), 100-116, 2005.

Richter, F.: Physikalische Eigenschaften von Stählen und ihre Temperaturabhängigkeit,

Stahleisen Sonderberichte Heft 10, Verlag Stahleisen, Düsseldorf, Germany, 1983

Schneider, R.: „Untersuchungen zum zeitabhängigen mechanischen Materialverhalten von S460

im Brandfall“, Diss. , Darmstadt, Germany, 2011

EN 1363: Fire Resistance Tests - Part 1: General Requirements

[10] Korzen, M.; Magonette G. and Buchet P.: “Mechanical Loading of Columns in Fire Tests by

Means of the Substructuring Method,” Zeitschrift für Angewandte Mathematik und Mechanik,

79, 617–618, 1999.

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]