[PPT] - Concrete members at Elevated Temperatures Peter Ansourian 1 , PowerPoint Presentation

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Partial Interaction Behaviour of Composite Steel- Concrete members at Elevated Temperatures

COST Action C26 Naples, Italy 16-18 September 2010

1 School of Civil Engineering, The University of Sydney, Sydney, Australia 2 University of Camerino, Ascoli Piceno, Italy

Peter Ansourian1, Gianluca Ranzi1 and Alessandro Zona2

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SUMMARY

Main aspects considered:

Composite steel/concrete beams at moderately elevated

temperatures

Numerical model
Elastic but degraded material properties

Acknowledgements

The second author was supported by the Australian Academy of Science and by the Australian Research Council under its Discovery Projects funding scheme.

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O X Y Z Az L Ax Ay

1 2

Typical composite steel-concrete beam and cross-section

O

X Y Yc 2 1

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UNPROTECTED STEEL

Composite beam (i.e. slab and steel joist joined by shear connectors) Combined tension and shear Undeformed shape

Based on full-scale tests, Zhao and Kruppa (1995) observed that uplift forces increase further during the cooling phase and reported pull-out failures in the mechanical devices used for the shear connection

Non-composite beam (i.e. slab and steel joist with no shear connectors) Typical thermal distribution at elevated temperatures Deformed shape Unprotected steel

Possible applicability of proposed model

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PROTECTED STEEL Brittle failures due to excessive deformation of the connectors have been reported by Zhao and Kruppa (1995).

Composite beam (i.e. slab and steel joist joined by shear connectors) Typical thermal distribution at elevated temperatures Fire protection A A B B Section A-A Combined tension and shear Undeformed shape Deformed shape Section B-B

Possible applicability of proposed model

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Assumption: 2 beams are considered denoted by α = 1,2. The two beams are coupled by an interface connection located at Y = Yc.

Z

w1 v1

1 2

w2 X Y y v2 Yc

Displacement and strain fields

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Assumption: cross-sections remain orthogonal to the deformed beam axis so that the rotation angle can directly be related to the axis displacements (Simo 1985):

e v 1 ' sin

e w 1 ' 1 cos

1 ' 1 '

2 2

w v Z e

where the prime denotes the derivative with respect to Z and the function

describes the axial strain.

Displacement and strain fields

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The above beam model is able to describe large deformations of the system. In this format it is however complicated to obtain a numerical solution. In the context of Civil Engineering, it is observed that the collapse of beams occurs when the maximum strains are very small (0.2 % to 1.0 %), while the maximum rotations of the cross-sections are about 1/20. These quantities can be considered moderately small. It is convenient to develop a simplified theory within this framework (Ranzi et al. 2010).

Displacement and strain fields

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Overview of displacement and strain fields Displacement and strain fields

dcy w1 v1

1 2

w2 dcz X Y y

Z

Yc v2 2 1

" ' 2 1 '

2 Yv

v w

z y

Yv w v A A u ) ' ( Under the assumptions of small strains and moderate rotations Generalised displacements

T

w v w v ] [

2 2 1 1

u

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Global and local balance conditions Global and local balance conditions The FE formulation is then derived based on this global balance condition.

L L S S L c c L S

Z Y X Z Y X Z Z Y X d d d ˆ d d d ˆ d ˆ d d d ˆ u t u b δ f

] ˆ , ˆ , ˆ [ u δc

Global balance condition

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Finite elements considered in the applications to validate proposed solutions (FE already available in the literature) Other finite elements considered in proposed applications 7dof FE element (GL-FI FE) Geometric linear finite element with full interaction

GL-FI FE

ve2 ve1 φe1 φe2 we1 we2 we3 ve2 ve1 φe1 φe2 we1 we2 we5 we6 we3 we4

GNL-FI FE

10dof FE element (GNL-FI FE) Geometric nonlinear finite element with full interaction

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and considering the partial interaction behaviour Other finite elements considered in proposed applications 22dof FE element (GL-PI FE) Geometric linear finite element with longitudinal and transverse partial interaction

ve22 ve21 φe21 φe22 we11 we21 we12 we22 we13 ve12 ve11 φe11 φe12 we15 we14 ve13 φe13 we23 we25 we24 ve23 φe23

GL-PI FE

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Proposed finite element

Finite element formulations

24dof FE element (GNL-PI FE) Geometric nonlinear finite element with longitudinal and transverse partial interaction Proposed finite element has been derived approximating the generalised displacements as:

e e z

z d N u ) ( ) (

ve22 ve21 φe21 φe22 ve12 ve11 φe11 φe12 we11 we12 we15 we16 we13 we14 ve13 φe13 ve23 we21 we22 we25 we26 we23 we24 φe23

GNL-PI FE

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Proposed applications

Applications

(A) Pinned member subjected to vertical uniformly distributed load applied to member 1 (at ambient temperature)

p1y Layer 2 Layer 1 Z

Y1

NOTE: supports located at level of centroid of composite cross-section unless noted otherwise

Case 1 Case 2

(B) Pinned member subjected to vertical uniformly distributed load applied to member 1 and thermal effects

Case 3

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Material properties

Applications

Linear-elastic material properties for two layers and longitudinal interface connection Nonlinear behaviour for transverse interface connection to better depict significant difference in stiffness between cases of vertical separation and penetration qL sL

For the vertical shear connection

L J E J E h A E A E k L

2 2 1 1 2 2 2 1 1

1 1

Non-dimentional parameter utilised in applications to depict level

f rigidity of both longitudinal and transverse interface connection:

In proposed simulations: longitudinal and transverse (when separating) interface connection rigidities are assumed to be identical

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 z /L

L =1 GNL-PI =1,10,50 GL-PI =10 GNL-PI

N

max

N

=50 GNL-PI GNL-PI GL-PI L L L

(e)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 z /L v

max 2

v

2

L =1

GNL-PI GL-PI

=10 L L =50

(a)

Application A: Pinned beam subjected to vertical load

Deflection of bottom layer Longitudinal slip Total axial force

1
0.5

0.5 1 0.2 0.4 0.6 0.8 1 L =10 =50 s

max L

s

L

z /L =1 GNL-PI GL-PI L L

(b)

0.2 0.4 0.6 0.8 1 10 100 N N

GNL-PI FE L

max

(f)

Total axial force

Case 1

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Application A: Pinned beam subjected to vertical load

Deflection of bottom layer Supports at level

f interface

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20 αL =10,20,50

GNL-PI GL-PI v

2 z = L/2

v 2 max

max

p

2 y

p

2 y

αL =10

z = L/2

αL =50 Case 2 0.2 0.4 0.6 0.8 1

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20

GNL-PI GL-PI

αL =10

max

p

2 y

p

2 y max

N

z = L/2

N

z = L/2

αL =1 αL =5,10,20,50 αL =50 Case 2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20,50

GNL-PI GL-PI v

2 z = L/2

v 2 max

max

p

2 y

p

2 y

αL =10

z = L/2

αL =1 αL =20 αL =5 αL =50 αL =10 0.2 0.4 0.6 0.8 1

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20,50

GNL-PI GL-PI

max

p

2y

p

2 y max

N

z = L/2

N

z = L/2

αL =1 αL =5,10,20,50 αL =10

Axial force Supports at level

f centroid of

cross-section

Case 2

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Application A: Pinned beam subjected to vertical load

Deflection of bottom layer Supports at level

f interface

Axial force Supports at level

f centroid of

cross-section

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20 αL =50

GNL-PI GL-PI v

2 z = L/2

v 2 max

max

p

2 y

p

2 y

αL =10

z = L/2

Case 3 0.2 0.4 0.6 0.8 1

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20,50 αL =1,5,10,20,50

GNL-PI GL-PI

αL =10

max

p

2 y

p

2 y max

N

z = L/2

N

z = L/2

Case 3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20 αL =50

GNL-PI GL-PI v

2 z = L/2

v 2 max

max

p

2 y

p

2 y

αL =10

z = L/2

0.2 0.4 0.6 0.8 1

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 αL =1 αL =5 αL =20 αL =50

GNL-PI GL-PI

αL =10

max

p

2 y

p

2 y max

N

z = L/2

N

z = L/2

Case 3

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Application B: Pinned beam subjected to UDL & thermal effects

MATERIAL PROPERTIES AND THEIR DEGRADATION WITH TEMPERATURE Retention function fDm(T ) quantifies the elastic properties for material m after they degrade at temperature T: m = c for concrete m = r for reinforcement and m = s for the steel joist related to properties at the reference temperature T0 = 20ºC. The constitutive relationship for the 3 materials is: σm =Em(εm- εTm ) where: Em = E0m fDm (T ) E0m = modulus at the reference temperature T0 = 20ºC εTm = thermal strain due to ∆T in material m

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Application B: Pinned beam subjected to UDL & thermal effects

Similar material representation is used for the interface shear connection

qz = fDsc.z (T )k0.zsz = kT.zsz

LONGITUDINAL SHEAR CONNECTION

where kT.z = longitudinal shear connection stiffness degraded according to an appropriate retention function fDsc.z.

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Application B: Pinned beam subjected to UDL & thermal effects

The stiffness of the transverse shear connection varies significantly for the cases of uplift (v2 – v1) > 0, and of bearing between the two layers (v2 – v1)

0. This bilinear representation takes the form:

y y T y y y Dsc y

s k s k T f q

. . . 1 2 1 2 . 1 2 1 2 .

v v v v k v v v v k

yb T yu T

where qy is the transverse force per unit length, fDsc.y(T) is the stiffness retention function for the transverse connection which relates the stiffness at the reference and elevated temperatures, i.e. , k0.y and kT.y

respectively. In particular, kT.yu and kT.yb are the transverse connection

stiffness for the cases of uplift and bearing respectively for which, realistically, kT.yb >> kT.yu.

TRANSVERSE SHEAR CONNECTION

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Application B: Pinned beam subjected to UDL & thermal effects

The constitutive models are assumed valid at service conditions and at relatively low temperatures, being applicable for stress levels in the steel less than its yield strength, and for compressive (tensile) stresses in the concrete less than about one half of its compressive (tensile) strength (Gilbert & Ranzi 2010). When the calculated stresses are outside this range or temperatures become relevant, the results nevertheless may still have a qualitative significance. REMARKS ON THE ADOPTED MATERIAL PROPERTIES

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Application B: Pinned beam subjected to UDL & thermal effects

Without loss of generality, the retention functions are here expressed by cubic polynomials. The generic retention function for material m can be written as where cim (i = 0,…,3) are appropriate coefficients for material m. Due to the complex nature of the degradation at increasing temperatures, it is prohibitive to establish one set of retention coefficients cim capable of depicting the degradation for a wide range of temperatures.

3 3 2 2 1

T c T c T c c f

m m m m Dm

RETENTION FUNCTIONS

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Application B: Pinned beam subjected to UDL & thermal effects

2 3 1 1 10 11 12 13 2 3 2 1 2 20 21 22 23 2 3 1 1 2 3

.

m D m m m m m m m D m m m m m Dm n m Dnm n m n m n m n m

T T f c c T c T c T T T T f c c T c T c T f T T f c c T c T c T The temperature domain is sub-divided into n domains and appropriate retention coefficients cjim (j = 1,..,n; i = 0,…,3) have been determined based on available EC guidelines:

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Application B: Pinned beam subjected to UDL & thermal effects

The thermal expansion coefficients have been determined based

n Eurocode 4 (2004), expressed as cubic polynomials to depict

the variation for material m at various temperature sub-domains:

2 3 1 10 11 12 13 1 2 3 2 1 2 20 21 22 23 2 3 1 1 2 3 T m m m m m m m m T m m m m m Tm n m Tnm n m n m n m n m

a a T a T a T T T T T T a a T a T a T T T T a a T a T a T

in which T = T – T0 EXPANSION COEFFICIENTS

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Application B: Pinned beam subjected to UDL & thermal effects

Last numerical example is carried

ut

using the geometry and cross-section

f a composite steel-concrete

beam used in a full-scale long-term test at USyd. Cross-sectional properties For illustrative purposes and without any loss of generality, a constant temperature distribution has been assumed to take place in the steel joist and raised with time. A UDL has been applied to provide a nominal contribution for the self-weight of the member. The beam is pinned at both its ends and supported at the level of its centroid. Temperature profile

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Application B: Pinned beam subjected to UDL & thermal effects

Presented results provided for two levels of shear connection:

SC-high = high level of shear connection stiffness
SC-low = low level of shear connection stiffness

Shear connection rigidities used in proposed simulations For simplicity, same shear connection stiffness has been adopted in the proposed simulations for the longitudinal and transverse (separation) shear connection.

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28 100 200 300 400 500 Temp

GNL-FI GNL-PI (with SC-high)

v/vref 0.5 1.0

GL-FI GL-PI (with SC-high)

100 200 300 400 500 Temp N/Nref

1.0
0.5

Application B: Pinned beam subjected to UDL & thermal effects

Validation against results obtained for high levels of shear connection (equivalent to FSI) and those calculated using FSI models.

0.5

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GNL-PI (with SC-low) GNL-PI (with SC-high)

v2 / v2.ref

GL-PI (with SC-low) GL-PI (with SC-high) Application B: Pinned beam subjected to UDL & thermal effects

Partial interaction analysis using two levels of shear connection.

100 200 300 400 500 Temp 100 200 300 400 500 Temp 0.5 1.0 N/Nref

1.0
0.5

0.5

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Presented work includes:

A numerical model with both partial interaction and geometric

nonlinearities to account for thermal effects

Weak form of the model
New finite element formulation
Applications

Conclusions

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