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

Partial Interaction Behaviour of Composite Steel- Concrete members at Elevated Temperatures Peter Ansourian 1 , Gianluca Ranzi 1 and Alessandro Zona 2 1 School of Civil Engineering, The University of Sydney, Sydney, Australia 2 University of Camerino, Ascoli Piceno, Italy COST Action C26 Naples, Italy 16-18 September 2010 1

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.

Typical composite steel-concrete beam and cross-section O Y c X 1 2 Z 1 A z O 2 Y X A x A y L Y 3

Possible applicability of proposed model UNPROTECTED STEEL Combined tension and Deformed Undeformed shear shape shape Unprotected steel Typical thermal distribution Non-composite beam Composite beam at elevated temperatures (i.e. slab and steel joist with no (i.e. slab and steel joist joined shear connectors) by shear connectors) 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

Possible applicability of proposed model PROTECTED STEEL Undeformed Combined B A shape tension and shear B Section B-B Fire protection Deformed shape Section A-A A Composite beam Typical thermal distribution at elevated (i.e. slab and steel joist joined by shear temperatures connectors) Brittle failures due to excessive deformation of the connectors have been reported by Zhao and Kruppa (1995). 5

Displacement and strain fields Assumption: 2 beams are considered denoted by α = 1,2. The two beams are coupled by an interface connection located at Y = Y c . 0 Z X y Y c w 1 v 1 w 2 v 2 1 Y 2 6

Displacement and strain fields 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): 1 w ' v ' cos sin 1 e 1 e where the prime denotes the derivative with respect to Z and the function 2 2 e Z v ' 1 w ' 1 describes the axial strain. 7

Displacement and strain fields 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). 8

Displacement and strain fields Overview of displacement and strain fields Under the assumptions of small strains and moderate rotations u A A v ( w Yv ' ) y z 1 2 Yv w ' v ' " 2 0 X Z y Y c w 1 1 v 1 w 2 2 Y v 2 1 d c z 2 d c y T Generalised displacements u [ v w v w ] 0 1 1 2 2 9

Global and local balance conditions Global and local balance conditions Global balance condition ˆ ˆ ˆ ˆ δ d X d Y d Z f d Z b u d X d Y d Z t u d X d Y d Z c c L S L L S L S ˆ ˆ ˆ δ c [ , , u ] The FE formulation is then derived based on this global balance condition. 10

Other finite elements considered in proposed applications Finite elements considered in the applications to validate proposed solutions (FE already available in the literature) GL-FI FE w e 2 w e 1 w e 3 φ e 1 φ e 2 v e 2 v e 1 7dof FE element (GL-FI FE) Geometric linear finite element with full interaction GNL-FI FE w e 2 w e 1 w e 3 w e 4 w e 5 w e 6 φ e 2 φ e 1 v e 2 v e 1 10dof FE element (GNL-FI FE) Geometric nonlinear finite element with full interaction 11

Other finite elements considered in proposed applications and considering the partial interaction behaviour φ e 13 w e 11 w e 14 w e 13 w e 15 w e 12 φ e 12 φ e 11 GL-PI FE v e 11 v e 12 v e 13 φ e 23 w e 21 w e 22 w e 24 w e 23 w e 25 φ e 22 φ e 21 v e 21 v e 22 v e 23 22dof FE element (GL-PI FE) Geometric linear finite element with longitudinal and transverse partial interaction 12

Finite element formulations Proposed finite element Proposed finite element has been derived approximating the generalised displacements as: u ( z ) N e z ( ) d 0 e w e 11 w e 13 w e 14 w e 15 w e 16 w e 12 φ e 11 φ e 12 GNL-PI FE v e 11 φ e 13 v e 12 v e 13 w e 21 w e 23 w e 24 w e 25 w e 26 w e 22 φ e 22 φ e 21 φ e 23 v e 21 v e 22 v e 23 24dof FE element (GNL-PI FE) Geometric nonlinear finite element with longitudinal and transverse partial interaction 13

Applications Proposed applications (A) Pinned member subjected to vertical uniformly distributed load applied to member 1 (at ambient temperature) Case 1 Z Y 1 p 1 y Layer 1 Case 2 Layer 2 Case 3 (B) Pinned member subjected to vertical uniformly distributed load applied to member 1 and thermal effects NOTE: supports located at level of centroid of composite cross-section unless noted otherwise 14

Applications Material properties Linear-elastic material properties for two layers and longitudinal interface connection q L Nonlinear behaviour for transverse interface connection to better depict significant difference s L in stiffness between cases of vertical separation and penetration In proposed simulations: longitudinal and For the vertical transverse (when separating) interface shear connection connection rigidities are assumed to be identical Non-dimentional parameter utilised in applications to depict level of rigidity of both longitudinal and transverse interface connection: 2 1 1 h L k L E A E A E J E J 1 2 1 1 2 2 1 2 15

Application A: Pinned beam subjected to vertical load Case 1 GL-PI s GNL-PI L GL-PI 1 s v L max GNL-PI Longitudinal slip 2 1 L =10 v 2 max L L =1 =10 0.5 0.8 L =50 z / L 0.6 0 0 0.2 0.4 0.6 0.8 1 0.4 -0.5 L =50 0.2 Deflection of bottom layer L =1 -1 0 (b) z / L 0 0.2 0.4 0.6 0.8 1 (a) GL-PI N GNL-PI N N max 1 L =1 GNL-PI N 0.8 max 0.8 GNL-PI FE Total axial force L =10 GNL-PI 0.6 L =50 GNL-PI 0.6 0.4 0.4 0.2 L 0.2 =1,10,50 GL-PI L Total axial force 0 0 16 z / L 0 0.2 0.4 0.6 0.8 1 1 10 100 (e) (f)

Application A: Pinned beam subjected to vertical load Case 2 Deflection of bottom layer Axial force GL-PI GL-PI p GNL-PI GNL-PI 2 y 1 1 p p max αL =5,10,20,50 αL =10,20,50 2 y 2 y p max αL =1 0.8 0.8 2 y αL =1 αL =50 αL =5 αL =1 0.6 0.6 αL =50 αL =5 αL =10 Supports at level αL =10 0.4 αL =20 0.4 of interface z = L/2 αL =20 z = L/2 N v 0.2 0.2 2 z = L/2 z = L/2 v 2 max N max 0 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Case 2 Case 2 GL-PI GL-PI p GNL-PI GNL-PI 1 2 y 1 p max αL =20,50 αL =1 αL =1 αL =20,50 2 y 0.8 αL =5 0.8 αL =1 αL =10 αL =1 αL =5,10,20,50 αL =10 0.6 0.6 αL =5 αL =5 Supports at level αL =10 0.4 0.4 of centroid of p z = L/2 N z = L/2 αL =20 v 2 y cross-section 0.2 αL =50 0.2 p 2 max z = L/2 2 y z = L/2 N v 2 max max 0 0 17 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Application A: Pinned beam subjected to vertical load Case 3 Deflection of bottom layer Axial force GL-PI GL-PI p GNL-PI GNL-PI 2 y 1 αL =1,5,10,20,50 1 αL =20 p max 2 y 0.8 0.8 αL =50 αL =1 αL =20,50 αL =1 0.6 0.6 αL =5 αL =5 Supports at level αL =10 of interface 0.4 0.4 p αL =10 z = L/2 N z = L/2 v 2 y 0.2 0.2 p 2 max z = L/2 2 y z = L/2 N v 2 max max 0 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Case 3 Case 3 GL-PI GL-PI p GNL-PI GNL-PI 1 2 y 1 p αL =20 max αL =50 2 y 0.8 0.8 αL =50 αL =1 αL =20 αL =1 0.6 0.6 αL =5 Supports at level αL =5 of centroid of 0.4 0.4 αL =10 p cross-section z = L/2 αL =10 N z = L/2 2 y v 0.2 0.2 p 2 max z = L/2 2 y N z = L/2 v 2 max max 0 0 18 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Application B: Pinned beam subjected to UDL & thermal effects MATERIAL PROPERTIES AND THEIR DEGRADATION WITH TEMPERATURE Retention function f Dm ( 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 T 0 = 20ºC. The constitutive relationship for the 3 materials is: σ m =E m ( ε m - ε Tm ) where: E m = E 0 m f Dm ( T ) E 0 m = modulus at the reference temperature T 0 = 20ºC ε Tm = thermal strain due to ∆T in material m 19

Application B: Pinned beam subjected to UDL & thermal effects Similar material representation is used for the interface shear connection q z = f Dsc.z ( T ) k 0.z s z = k T.z s z LONGITUDINAL SHEAR CONNECTION where k T.z = longitudinal shear connection stiffness degraded according to an appropriate retention function f Dsc.z . 20