Mergos PE, Beyer K - City, University of London · PDF fileModelling Shear-Flexure Interaction in Equivalent Frame Models of Slender RC Walls Mergos PE, Beyer K City University London

Modelling Shear-Flexure Interaction in

Equivalent Frame Models of Slender RC Walls

Mergos PE, Beyer K

City University London

Research Centre for Civil Engineering Structures

Importance of RC walls

• Reinforced concrete (RC) walls are widely used to provide lateral stiffness and

strength in medium to tall buildings

Advantages

•Inherently stiff limit inter-storey drifts and damage to non-structural

components

•Prevent soft-storey mechanisms protect against collapse

Capacity design of slender RC walls

1. Designed to form a (ductile) flexural mechanism at the base when loaded

beyond the elastic limit

2. They are provided with adequate shear resistance to prevent brittle shear

failures

Introduction

Shear wall – frame interaction

Wall-frame interaction

Before flexural yielding

• Shear deformations important because:

1. Significant depth of RC walls

2. Shear cracking

After flexural yielding

Experimental evidence, which goes back as far as

the 1970s (Wang et al., 1975; Oesterle et al.,

1976; Valenas et al., 1979), has shown that shear

deformations of RC walls continue to increase

after flexural yielding.

Shear deformations

Shear displacement (mm)S

he

ar

(kN

)

Shear to flexural displacement ratio Δs/Δf(Beyer et al. 2011)

Results quasi-static cyclic tests on slender, capacity-designed cantilever RC walls

suggest that the ratio of shear to flexural displacement Δs/Δf remains

approximately constant over the entire range of applied displacement ductilities.

Shear deformations

Δs/Δf paradox?

Shear deformations

Hn

LwV

V

γ

GAoGA1

flexural yielding

V-γ in hinge regions

V

φ

ΕΙ1

flexural yielding

M-φ in hinge regions

ΕΙο

stirrup yielding

Shear-flexure interaction

Cantilever RC wall


Truss analogy model Mohr’s circle at the centerline of wall

Strain profile at wall base section

εm=mean axial strainεh=horizontal strainεd=axial strain in compression strutγ=shear strainβ=cracking angle against element axis

Flexure dominated walls: εh and εd << εm

Beyer, Dazio and Priestley (2011)


Crack pattern Actual curvature profile

Plastic hinge length method

Assumption:A constant curvature φ and axial strain εm over the plastic hinge length Lph is assumed

•Shear displacement:

•Flexural displacement:

•Ratio Δs/Δf:

Plastic curvature profile

Prediction

Δs/Δf

Ex

pe

rim

en

t

Beyer, Dazio and Priestley (2011)

RC walls finite element models

• Inelastic response can be well captured by shell elements with advanced analytical methodologies accounting for the biaxial in-plane stress state in RC elements (e.g. MCFT by Vecchio and Collins 1986).

• Other models (e.g Wallace et al.) use macro-elements in which different elements are assigned a specific load-carrying mechanism (i.e. axial forces, moments and shears).

• In engineering practice, however, beam-column elements are typically employed.

– The vast majority of beam-column elements model only flexural and axial response.

– Beam-column models with shear-flexure interaction are very limited.

– Typically, a constant shear rigidity GA is assigned. This follows the misconception that GA is not influenced by flexural deformations.

Finite Element Model

RC Member

Finite Element

Sub-elements

Bond-Slip

(+)

(+)

F=Ffl + Fsh + Fsl

Shear

Flexural

(Mergos & Kappos 2012)

Flexural subelement

ΕΙΑ

ΕΙο

ΕΙΒ

A Β

MΑ

MB

MyB

MyA

L

αΑL αΒL (1- αΑ-αΒ)L

Elastic zone

Inelastic zones

ΜΑ

ΜyΑ

ΜB

ΜyB

dxxEI

xmxmf

L

BAfl

ij

0

)(

)()( 12

ij o A A B B

o

Lf c c c

EI Flexibility matrix

(Soleimani et al. 1979)

αAL

L

αBL

co,cA, cB functions of αA and αB

Flexural hysteretic response

Monotonic M-φ Fiber analysis & bilinearization

(Soleimani et al. 1979)

Hysteretic M-φ Mild stiffness degradation

Stiffness distribution in inelastic zones?

Shear subelement

(Mergos & Kappos 2012)

ΕΙΑ

ΕΙο

ΕΙΒ

A Β

MΑ

MB

MyB

MyA

L

αΑL αΒL (1- αΑ-αΒ)L

Elastic zone

Inelastic zones

ΜΑ

ΜyΑ

ΜB

ΜyB

•Flexibility matrix dxxGA

xvxvf

L

BAsh

ij

0

)(

)()(

V-γ without

interaction

V-γ with

interaction

•Moment diagram

•Flexural sub-element

•Shear sub-element

αAsL αBsL

L

GAA

GAM

GAB

Shear hysteretic response

Severe hysteretic degradation

Monotonic V- γ Hysteretic V-γ

• Uncracked shear stiffness GAo

• Cracked shear stiffness GA1 determined by truss

analogy approach (Park and Paulay, 1975) with

modification factors by Mergos and Kappos (2012)

• Alternatively, GAeff=Vy/γy (γy=0.0013 EC8-Part 3)(Park & Paulay 1975)

(Ozcebe & Saatcioglu 1989)

Modelling shear-flexure interaction

Aim: Establish GA2 in plastic hinge regions

General loading conditions:

Strain – curvature (γ-φ) relationship

Written in incremental form

Definition of tangent shear stiffness

Assumption: c=neutral axis depth≈constantStrain profile

Lw

RC wallHn


General loading conditions:

ΔV GA2 Need for iterative scheme (typically converges very fast)


Cantilevers and walls with constant shear span lengths Hn

General formula:

ΔV= ΔM/Hn

ΔM/ Δφ=EI1

Lw

Monotonic V- γ

Monotonic M-φ

After flexural yielding


Cantilevers and RC walls with constant shear span lengths Hn

Increment of shear displacements:

Uniform GA3

V

γ

GA3

flexural yielding

Uniform V-γ

GA1

Model Validation

• Model fully implemented in IDARC2D (State University of New York at Buffalo)

• The numerical results were compared with experimental results obtained from a

quasi-static cyclic test of a rectangular RC wall (WSH3 in Dazio et al., 2009).

• The test unit was 2.00m long and 0.15m wide and had a shear span of 4.56m

• During cyclic loading, the specimen was subjected to a constant axial load of

686 kN

Side view

Total displacement (mm)

Sh

ea

r (k

N)

Reinforcement layout

Model Validation

Shear displacement (mm)

Δs/Δ

f

Sh

ea

r (k

N)

Sh

ea

r (k

N)

Total displacement (mm)

Curvature (rad/km) Shear strain

Mo

me

nt

(kN

m)

Model Validation

Flexural stiffness values

EIg=3.5·1012 kNm2

EI0/EIg=22%

EI1/EIg=0.16%

Monotonic M-φ

Shear stiffness values

GAg=GAo=3.4·106 kN

GA1/GAg=13.9%

GA2/GAg=0.08%

GA3/GAg=0.8%

Monotonic V- γ

Model Validation

Pushover analysis of a single RC wall with varying point of contra-flexure

• The previous RC wall is assumed to have twice its original height and be rotationally restrained at top

• Base section M-φ same as previously, but top section M-φ infinitely elastic (never yields)

• After yielding, point of contra-flexure moves towards the wall base

• The wall is subjected to pushover analysis up to 3% relative drift

Moment and shear distributions at two subsequent steps

Model Validation


Moment and shear distributions at two subsequent steps

Drift (%)

Hn/H

tot

RC wall

Model Validation


Moment and shear distributions at two subsequent stepsRC wall

Drift (%)Base curvature (1/m)

Sh

ea

r str

ain

Δs/Δ

f

Constant Hn: Proposed:

Model Implementation

Pushover analysis of interconnected RC walls with different lengths

• 8-storey wall building (24m high).

• 2 walls with different length (6m and 4m respectively)

• Walls are coupled via RC slabs assumed infinitely stiff

• Aim: Examine variation of wall shear forces


Pushover analysis of interconnected RC walls with different lengths

Top displacement (mm) Top displacement (mm)

Top displacement (mm)

Sh

ea

r (k

N)

Sh

ea

r (k

N)

Sh

ea

r (k

N)

Building

Long wall Short wall


Pushover and time-history analysis of a wall-frame structure

• 2-span, 10-storey wall-frame building (30m high and 16m long).

• Designed according to EC8 for ductility class “M” (Penelis and Kappos 1997)

• 1 wall, 4m long

• End columns 0.4X0.4m

• Aim: Examine shear-flexure interaction effect of RC wall


Pushover analysis

Top Drift (%)

Vb

ase/W

@ 2%

Storey drift (%)

Sto

rey

Pushover analysis


Columns @ 2%

Curvature ductility μφ

Sto

rey


Sto

rey

Beams @ 2%

Time history AnalysisEl-Centro 1940 N-S


Time (sec)

To

p d

isp

lace

me

nt

(mm

)

Time (sec)

Gro

un

d s

tore

y

dis

pla

ce

me

nt

(mm

)



Columns


Sto

rey

Storey drifts

Sto

rey



Columns


Sto

rey

Storey drifts

Sto

rey


Simplified estimation of ground storey drift due to shear deformations

For wall-frame systems without beams:

Conclusions

•Experimental evidence has shown that shear deformations of RC walls

continue to increase after flexural yielding. This is caused by shear-flexure

interaction effect.

•Beam-column elements used in practice to model RC walls ignore shear-

flexure interaction by assuming GA remains constant after flexural yielding

•A new beam-column element that accounts for shear-flexure interaction has

been developed and implemented in IDARC

•Validation studies have proven the accuracy of the proposed model

•Shear-flexure interaction is particularly important for predicting column

demands at the ground storey of tall wall-frame buildings

References

•Dazio, A., Beyer, K. and Bachmann, H. (2009). Quasi-static cyclic tests

and plastic hinge analysis of RC structural walls. Engineering

Structures 31, 1556-1571.

•Beyer, K., Dazio, A. and Priestley, M.J.N. (2011). Shear deformations of

slender RC walls under seismic loading. ACI Structural Journal 108:2,

167-177.

•Mergos, P.E. and Kappos, A.J. (2012). A gradual spread inelasticity

model for R/C beam-columns accounting for flexure, shear and

anchorage slip. Engineering Structures 44:1, 94-106.

•Mergos, P.E. and Beyer, K. (2014). Modelling shear-flexure interaction in

equivalent frame models of slender RC walls. Structural Design of Tall

and Special Buildings 23:15, 1171-1189.

Thank you for your kind attention…

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Mergos PE, Beyer K - City, University of London · PDF fileModelling Shear-Flexure Interaction in Equivalent Frame Models of Slender RC Walls Mergos PE, Beyer K City University London