Computational limit state analysis of reinforced concrete structures

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1 Computational limit state analysis of reinforced concrete structures Nunziante Valoroso Università di Napoli Parthenope

2 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

3 Motivation I Nonlinear static analysis is becoming increasingly popular in practical applications for assessing the performance of Reinforced Concrete (RC) structures under seismic load action. In Civil Engineering literature it is also known as Pushover Analysis, alternative to nonlinear transient dynamic analysis for large-scale engineering structures. For a moderately wide class of structural systems the method can predict the seismic force and deformation demands at an affordable computational cost.

4 Motivation II Less accurate compared to a fully nonlinear dynamic analysis. Pushover analysis can provide valuable information on the structural response provided that the inelastic behavior of all the structural elements is consistently described. In this contet the correct description of material behaviour is of paramount importance in order to capture the structural mechanisms.

5 Motivation IV How much refined should constitutive laws be to effectively carry out limit state analyses computations for real large-scale structures?

6 Answer 1

7 Figure: The multifiber concept

8 Answer 2

9 0.13F F Figure: Four-point shear test. Model problem

10 Figure: 4-point test. FE mesh and damage distribution (Jirasek, 2001)

11 (a) (b) (c) (d) Figure: Discrete approaches to crack propagation: (a) remeshing; (b) no remeshing; (c) global enrichment; (d) local enrichment.

12 EUROPEAN STANDARD NORME EUROPÉENNE EUROPÄISCHE NORM EN December 2004 ICS ; Supersedes ENV :1991, ENV :1994, ENV :1994, ENV :1994, ENV :1994, ENV :1998 English version Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings Eurocode 2: Calcul des structures en béton - Partie 1-1 : Règles générales et règles pour les bâtiments Eurocode 2: Bemessung und konstruktion von Stahlbetonund Spannbetontragwerken - Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau This European Standard was approved by CEN on 16 April CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the Central Secretariat or to any CEN member. This European Standard eists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CEN member into its own language and notified to the Central Secretariat has the same status as the official versions. CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom. EUROPEAN COMMITTEE FOR STANDARDIZATION C O M I T É E U R O P É E N D E N O R M A LIS A T I O N EUROPÄIS C HES KOMITEE FÜR NORMUNG Management Centre: rue de Stassart, 36 B-1050 Brussels 2004 CEN All rights of eploitation in any form and by any means reserved worldwide for CEN national Members. Ref. No. EN :2004: E

13 fcd For structural analysis c Schematic c For section analysis Parabola-rectangle c Bi-linear fcm fck fck fcd 0,4 fcm tan = Ecm c1 ( ) 0,7 f cm 0,31 cu1 ( ) = c1 cu1 2,8 + 27[(98-f cm )/100] 4 f cm )/100] 4 for f ck 50 MPa otherwise 3.5 c c2 cu2 0 c fcd n c 1 1 for 0 c c2 c2 c fcd for c2 c cu2 n = 1,4 + 23,4 [(90- f ck )/100] 4 for f ck 50 MPa otherwise 2,0 c2 ( ) = 2,0 + 0,085(f ck -50) 0,53 c 0 c3 cu3 c3 ( ) = 1,75 + 0,55 [(f ck -50)/40] for f ck 50 MPa otherwise 1,75 cu3 ( ) =2,6+35[(90-f ck )/100] 4 for f ck 50 MPa otherwise 3,5 c for f ck 50 MPa otherwise 2,0 cu2 ( ) = 2, [(90-f ck )/100] 4 for f ck 50 MPa otherwise 3,5 Figure: Concrete stress-strain relations (3.1.5 and 3.1.7)

14 Idealised kfyk fyk kfyk/ s fyd = fyk/ s Design k = (f t /f y ) k fyd/ Es ud uk ud = 0.9 uk Figure: Idealized and design stress strain relations for reinforcing steel

15 Ch. 5.7 Nonlinear analysis: Nonlinear analysis may be used for both ULS and SLS, provided that equilibrium and compatibility are satisfied and an adequate non-linear behaviour for materials is assumed. The analysis may be first or second order.

16 Goal: develop a true engineering approach suitable for real-scale computations of RC structures. A smart quadrature technique is being used to avoid inaccurate and time-consuming computations. Essential is a robustness requirement: a viable approach can be confidently used to analyze full-scale structures based on a minimal set of material parameters.

17 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

18 Fz M M d z beam model shell model shell model beam model beam model shell model Figure: RC walls subject to concentrated forces and stress couples.

19 concrete layers steel layer t t b Figure: Layered slab with through-the-thickness strain and stress distribution.

20 Constitutive laws Steel H c(ε ε yc) + σ yc σ s(ε) = E ε H t(ε ε yt) + σ yt if ε < ε yc if ε yc ε ε yt if ε yt < ε ε yc σ σ yt E ε yt H t ε H c σ yc Concrete σ c(ε) = 0 if 0 < ε ) σ co (2 ε ε2 if ε co ε 0 ε co ε co σ co if ε < ε co σ σ co ε ε co

21 Frame sections: Fiber-free approach Rebars y r 3 N s = n b σ s[ε(s j )]A j ; M s = ( M sy, M s) t = n b j=1 j=1 σ s[ε(s j )]s j A j Concrete N c = σ c[ε(r)]dω; M c = ( M cy, M c) t = σ c[ε(r)]rdω Ω Ω r 6 r 4 s5,a 5 r 5 s 7,A 7 s 4,A 4 s 6,A 6 s 3,A 3 s 2,A 2 s 1,A 1 r 1 r 2 N c = n i=1 4 k=0 l i (ĝ n i ) Φ 0 i [σ (1) c (ε)] if g > tol g σ ( k) I k g k if g < tol g k! Φ 0 i [σ c (1) (ε)] = σ (2) c (ε i+1 ) σ (2) c (ε i ) ε i+1 ε i if ε i+1 ε i > tol ε σ (1) c ( ε i ) + σ( 1) c ( ε i ) (ε i+1 ε i ) 2 if ε i+1 ε i < tol ε 24

22 RC shells computations II Gauss points RC section Gauss point t b l s l t s l b Figure: A typical finite element and its partition into quadrature subcells.

23 RC shells computations II ǫ = ε z (ˆ 0 ) = ε b (ˆ 0 ) for b {1, 2} (1) χ = ε z ε b = for b = 1, s = 2 s 0 ˆ = y 0 ˆ + ε (2) b = for b = 2, s = 1 s 0 ˆ χ y = ε z = ε b = κ b (ˆ 0 ) for b {1, 2} (3) 0 ˆ 3 0 ˆ

24 RC shells computations III N b = 1 l s l s N b d s = N l y (4) M bs = 1 1 l y N b s d s = l s l s 1 l y Ω Ω σ z y dω = M for b = 1, s = 2 l y M σ z y dω = for b = 2, s = 1 l y (5) M b = 1 l s l s M b d s = M y l y (6)

25 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

26 Planar wall-frame structure I p p A Beams p p y B 0.30 Columns Wall 3.00 Figure: Wall-frame reinforced concrete structure.

27 Planar wall-frame structure II Limit states ULS (bars) ELS (bars) ELS (conc) ULS (conc) (a) (b) Figure: Deformed shape and limit states. Beam VS shell

28 Planar wall-frame structure III χ [m 1 ] Beam A Shell model Beam model [m] χ [m 1 ] Beam B Shell model Beam model [m] Figure: Curvatures distribution along beams A and B. Beam VS shell.

29 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

30 Cervenka-Gerstle panel I Figure: Cervenka-Gerstle benchmark (1971).

31 Cervenka-Gerstle panel II Figure: Cervenka-Gerstle panel. Crack and damage patterns

32 Cervenka-Gerstle panel III 0.12 RC Shell [Cervenka and Gerstle, 1971] Load [MN] Eperiment by Cervenka and Gerstle FE Analysis (Parabola rectangle) FE Analysis (Bilinear with softening) Displacement [m] Figure: Global load-deflection responses

33 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

34 TW2 wall I DETAIL A φ DETAIL B φ6.4 / φ4.75 / φ9.5 Typ φ4.75 / DETAIL A φ4.75 / φ6.4 / φ4.75 / φ9.5 DETAIL C DETAIL C DETAIL B Figure: Thomsen-Wallace shear walls (1994). NEES database.

35 TW2 wall II Lateral load [kn] Eperimental curve Computed curve Top displacement [m] Figure: Load-deflection curve for TW2 wall.

36 TW2 wall III Limit states ULS (bars) ELS (bars) ELS (conc) ULS (conc) Figure: Moment-curvature and limit states for TW2 wall

37 Outline 1 Introduction Motivation 2 Background RC structures computation 3 Applications Zero One Two Three 4 Closure

38 RC building I Top view F 1250 y Front view F Columns F 30 F Beams F z A B C Shear walls Figure: Shear walled building: structural plans.

39 RC building II Base shear [kn] T=2.09 T=2.00 T=2.30 T= Top displacement [m] Figure: Shear walled building. Base shear VS top displacement.

40 RC building III Shell model Beam model Limit states ULS (bars) ELS (bars) ELS (conc) ULS (conc) (a) Alignment at y = 0.00 m (b) Alignment at y = m Figure: Shear walled building. Limit states at 0.20 m top displacement.

41 RC building IV Shell model Beam model Limit states ULS (bars) ELS (bars) ELS (conc) ULS (conc) (a) Alignment at y = 0.00 m (b) Alignment at y = m Figure: Shear walled building. Limit states at 0.75 m top displacement.

42 RC building V Section A Shell model Beam model ε z [m] Section B Shell model Beam model ε z [m] Section C Shell model Beam model ε z [m] Figure: Distribution of vertical strain along sections A,B,C. Beam VS shell.

43 Closure I A reinforced shell element is presented for the nonlinear static analysis of concrete structures containing shear walls. Stress state is integrated in closed-form (fiber-free). The model has been validated against eperimental results. Effects of localized actions on shear walls due to interaction with nearby framed structures are well captured. Numerical results show the ability of the approach to analyze full-scale structures at a reduced computational cost. Many etensions possible...

44 Shear walled building Gauss points Base shear [kn] Gauss point RC section t 4000 l t s b l s 2000 l b Top displacement [m] Thanks for attention nunziante.valoroso@uniparthenope.it Reference: N. Valoroso et al., Limit state analysis of reinforced shear walls, Engineering Structures,

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