Computational limit state analysis of reinforced concrete ...€¦ · Computational limit state...

Computational limit state analysisof reinforced concrete structures

Nunziante [email protected]

Università di Napoli Parthenope

N. Valoroso (Università Napoli Parthenope) RC structures

Outline

1 IntroductionMotivation

2 BackgroundRC structures computation

3 ApplicationsZeroOneTwoThree

4 Closure


Motivation I

Nonlinear static analysis is becoming increasingly popular in practicalapplications 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-scaleengineering structures.

For a moderately wide class of structural systems the method can predictthe seismic force and deformation demands at an affordablecomputational cost.


Motivation II

Less accurate compared to a fully nonlinear dynamic analysis.

Pushover analysis can provide valuable information on the structuralresponse provided that the inelastic behavior of all the structuralelements is consistently described.

In this context the correct description of material behaviour is ofparamount importance in order to capture the structural mechanisms.


Motivation IV

How much refined should consti-tutive laws be to effectively carryout limit state analyses compu-tations for real large-scale struc-tures?


Answer 1


Figure: The multifiber concept


Answer 2


F0.13F

203 397 397 203

82

224

61 61

Figure: Four-point shear test. Model problem


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


(a) (b)

(c) (d)

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


EUROPEAN STANDARD

NORME EUROPÉENNE

EUROPÄISCHE NORM

EN 1992-1-1

December 2004

ICS 91.010.30; 91.080.40 Supersedes ENV 1992-1-1:1991, ENV 1992-1-3:1994,ENV 1992-1-4:1994, ENV 1992-1-5:1994, ENV 1992-1-

6:1994, ENV 1992-3:1998

English version

Eurocode 2: Design of concrete structures - Part 1-1: Generalrules 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 Stahlbeton-und Spannbetontragwerken - Teil 1-1: AllgemeineBemessungsregeln und Regeln für den Hochbau

This European Standard was approved by CEN on 16 April 2004.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this EuropeanStandard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such nationalstandards may be obtained on application to the Central Secretariat or to any CEN member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by translationunder the responsibility of a CEN member into its own language and notified to the Central Secretariat has the same status as the officialversions.

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, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia,Slovenia, Spain, Sweden, Switzerland and United Kingdom.

EUROPEAN COMMITTEE FOR STANDARDIZATIONC O M I T É E U R O P É E N D E N O R M A LI S A T I O NEUR OP ÄIS C HES KOM ITEE FÜR NOR M UNG

Management Centre: rue de Stassart, 36 B-1050 Brussels

© 2004 CEN All rights of exploitation in any form and by any means reservedworldwide for CEN national Members.

Ref. No. EN 1992-1-1:2004: E


fcd

c2

c

! cu2 c0

fck

For section analysis

“Parabola-rectangle”

c3

cu30

fcd

!c

c

fck

“Bi-linear”

fcm

0,4 fcm

c1

c

cu1 c

tan = Ecm

For structural analysis

“Schematic”

c1 (!"!!) #$0,7 fcm

0,31

cu1 (!"!!

) =

2,8 + 27[(98-fcm)/100]4 fcm)/100]4

for fck 50 MPa otherwise 3.5

c2 (!"!!) = 2,0 + 0,085(fck-50)0,53

for fck 50 MPa otherwise 2,0

cu2 (!"!!) = 2,6 + 35 [(90-fck)/100]4


n = 1,4 + 23,4 [(90- fck)/100]4


f

n

cc cd c c2

c2

1 1 for 0

% &' () *# + + , -. /) *0 12 3

f forc cd c2 c cu2 # , ,

c3 (!"!!) = 1,75 + 0,55 [(fck-50)/40]


cu3 (!"!!

) =2,6+35[(90-fck)/100]4


Figure: Concrete stress-strain relations (3.1.5 and 3.1.7)


ud

!

fyd/"Es

fyk

kfyk

fyd = fyk/ s

kfyk/ s

Idealised

Design

uk

ud= 0.9 uk

k = (ft/fy)k

Figure: Idealized and design stress strain relations for reinforcing steel


Ch. 5.7 Nonlinear analysis:Nonlinear analysis may be usedfor both ULS and SLS, providedthat equilibrium and compatibil-ity are satisfied and an adequatenon-linear behaviour for materi-als is assumed. The analysismay be first or second order.


Goal: develop a true engineering approach suitable for real-scalecomputations of RC structures.

A smart quadrature technique is being used to avoid inaccurate andtime-consuming computations.

Essential is a robustness requirement: a viable approach can beconfidently used to analyze full-scale structures based on a minimal setof material parameters.


Outline




4 Closure


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



x

x

x

x

x

Fz

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

xdz

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx













beam modelshell model




Mx

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x







x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x




Mx



Figure: RC walls subject to concentrated forces and stress couples.


concrete layers

steel layer

t

b

t

Figure: Layered slab with through-the-thickness strain and stress distribution.


Constitutive laws

Steel

σs(ε) =

Hc(ε− εyc) + σyc if ε < εyc

E ε if εyc ≤ ε ≤ εytHt(ε− εyt) + σyt if εyt < ε

σ

E

Ht

σyt

σyc

Hc

εεyt

εyc

Concrete

σc(ε) =

0 if 0 < ε

σco

εco

(2 ε− ε2

εco

)if εco ≤ ε ≤ 0

σco if ε < εco

σco

εcoε

σ

N. Valoroso (Universita Napoli Parthenope) RC structures

Frame sections: Fiber-free approach

Rebars

Ns =

nb∑j=1

σs [ε(sj)]Aj ; M⊥s = (−Msy ,Msx)t =

nb∑j=1

σs [ε(sj)]sjAj

Concrete

Nc =

∫Ω

σc [ε(r)]dΩ; M⊥c = (−Mcy ,Mcx)t =

∫Ω

σc [ε(r)]rdΩr1

y

x

r2

r6

r3

r4

r5

s1,A1

s2,A2

s3,A3

s4,A4s5,A5

s6,A6

s7,A7

Nc =

n∑i=1

li (g · ni ) Φ0i [σ(1)

c (ε)] if |g| > tolg

4∑k=0

σ(−k)

k!Ik · g⊗k if |g| < tolg

Φ0i [σ(1)

c (ε)] =

σ

(2)c (εi+1)− σ(2)

c (εi )

εi+1 − εiif |εi+1 − εi | > tolε

σ(1)c (εi ) +

σ(−1)c (εi )

24(εi+1 − εi )2 if |εi+1 − εi | < tolε

N. Valoroso (Universita Napoli Parthenope) RC structures

RC shells computations II

Gauss points

Gauss pointxt

xb

xs

lb

ls

lt

RC section

Figure: A typical finite element and its partition into quadrature subcells.


ǫ = εz(x0) = εb(x

0) for b ∈ 1, 2 (1)

χx =∂εz

∂y

∣∣∣∣x0

=

−∂εb

∂xs

∣∣∣∣x0

= for b = 1, s = 2

+∂εb

∂xs

∣∣∣∣x0

= for b = 2, s = 1

(2)

− χy =∂εz

∂x

∣∣∣∣x0

=∂εb

∂x3

∣∣∣∣x0

= κb(x0) for b ∈ 1, 2 (3)


RC shells computations II

Nb =1ls

∫

ls

Nb dxs =Nly

(4)

Mbs =1ls

∫

lsNbxs dxs =

−1ly

∫

Ω

σzy dΩ = −Mx

lyfor b = 1, s = 2

1ly

∫

Ω

σzy dΩ =Mx

lyfor b = 2, s = 1

(5)

Mb =1ls

∫

lsMb dxs = −

My

ly(6)


RC shells computations III

Outline




4 Closure


Planar wall-frame structure I

5.50 3.00 5.50

3.00

3.00

0.30

0.50

Beams !"#!$%

x

y

Columns !"#!$%

0.50

0.30

Wall !"#$%&

0.30

3.00

p

A

B

p p

p

Figure: Wall-frame reinforced concrete structure.


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


Planar wall-frame structure III

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

−0.1

−0.05

0

0.05

0.1

0.15

x [m]

χ [m

−1 ]

Beam A

Shell modelBeam model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

−0.1

−0.05

0

0.05

0.1

0.15

x [m]

χ [m

−1 ]

Beam B


Figure: Curvatures distribution along beams A and B. Beam VS shell

.


Outline




4 Closure


Cervenka-Gerstle panel I

Figure: Cervenka-Gerstle benchmark (1971).


Cervenka-Gerstle panel II

Figure: Cervenka-Gerstle panel. Crack and damage patterns


Cervenka-Gerstle panel III

0 0.002 0.004 0.006 0.008 0.010

0.02

0.04

0.06

0.08

0.1

0.12RC−Shell [Cervenka and Gerstle, 1971]

Load

[MN

]

Displacement [m]

Experiment by Cervenka and GerstleFE−Analysis (Parabola−rectangle)FE−Analysis (Bilinear with softening)

Figure: Global load-deflection responses


Outline




4 Closure


TW2 wall I

1.2192

3x0.0508

0.019050.019050.0635

0.019050.019050.0635

3x0.1016

0.1016

1.2192

φ6.4 / 0.1905

φ6.4 / 0.1397

0.0508 φ6.4

φ4.75 / 0.1016

φ4.75 / 0.0381

φ4.75 / 0.03175

8φ9.5

3x0.0508 0.01905

0.0635

φ9.5 0.01905φ4.75 / 0.0762

DETAIL A

DETAIL B

DETAIL C

DETAIL A

DETAIL B

DETAIL C

Typ.

Figure: Thomsen-Wallace shear walls (1994). NEES database.


TW2 wall II

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

20

40

60

80

100

120

140

160

180

200

Late

ral l

oad

[kN

]

Top displacement [m]

Experimental curveComputed curve

Figure: Load-deflection curve for TW2 wall.


TW2 wall III

Limit states

ULS (bars)

ELS (bars)

ELS (conc)

ULS (conc)

Figure: Moment-curvature and limit states for TW2 wall


Outline




4 Closure


RC building I

540 200 340 340 340 200 54039

045

040

0

1250

750

F

320

320

320

60

30

220

30 !"#!$ !$#%& !"#!$

30

3065

Top view

Front view

Shear walls

60

Columns

Beams

520

x

y

540 200 340 340 340 200 540x

z

!"#!$%

&#!$%

A

B

C

F

F

F

F

Figure: Shear walled building: structural plans.


RC building II

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

2000

4000

6000

8000

10000

12000

14000

16000

Bas

e sh

ear

[kN

]


T=2.00

T=2.09

T=2.30

T=2.20

Figure: Shear walled building. Base shear VS top displacement.


RC building IIIShell model Beam model

Limit states

ULS (bars)

ELS (bars)

ELS (conc)

ULS (conc)

(a) Alignment at y = 0.00 m

(b) Alignment at y = 13.60 m

Figure: Shear walled building. Limit states at 0.20 m top displacement.


RC building IVShell model Beam model

Limit states

ULS (bars)

ELS (bars)

ELS (conc)

ULS (conc)

(a) Alignment at y = 0.00 m

(b) Alignment at y = 13.60 m

Figure: Shear walled building. Limit states at 0.75 m top displacement.


RC building V

14.5 15 15.5 16 16.5 17 17.5−1

−0.5

0

0.5

1x 10

−3

x [m]

ε z

Section A


14.5 15 15.5 16 16.5 17 17.5−1

−0.5

0

0.5

1x 10

−3

x [m]

ε z

Section B


14.5 15 15.5 16 16.5 17 17.5−0.1

0

0.1

0.2

0.3

x [m]

ε z

Section C


Figure: Distribution of vertical strain along sections A,B,C. Beam VS shell.


Closure I

A reinforced shell element is presented for the nonlinear static analysis ofconcrete structures containing shear walls.

Stress state is integrated in closed-form (fiber-free).

The model has been validated against experimental results.

Effects of localized actions on shear walls due to interaction with nearbyframed structures are well captured.

Numerical results show the ability of the approach to analyze full-scalestructures at a reduced computational cost.

Many extensions possible...


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

2000

4000

6000

8000

10000

12000

14000

16000Shear walled building

Bas

e sh

ear

[kN

]


Gauss points

Gauss pointxt

xb

xs

lb

ls

lt

RC section

Thanks for attention

[email protected]

Reference: N. Valoroso et al., Limit state analysis of reinforced shear walls, Engineering Structures, 2014


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