1

Reducing uncertainty in the assessment of

the masonry buildings by moving from

empirical to analytical drift capacity

models

Katrin Beyer, EPFL

http://eesd.epfl.ch/

The 42nd Risk, Hazard & Uncertainty Workshop, Hydra, Greece

June 2016

2

Collaborators

• Bastian Wilding, Dr. Sarah Petry, Marco Tondelli, Dr. Panos Mergos

Grants

• FP7-Programme for access to TREES laboratory of EUCENTRE

• Swiss National Science Foundation

• Federal Office for the Environment in Switzerland (OFEV)

• In-kind contributions by Morandi Frères SA

Acknowledgments

3

Drift capacity of unreinforced masonry walls

State of practice (e.g. Eurocode 8)

Empirical model

• Shear failure: du=0.4%

• Flexural failure: du=0.8% H0/L

H0=Shear span

L= Wall length

State of the art

Empirical models with further parameters

• Axial stress ratio

• Parameter describing moment profile H0/H

Shear failure Flexural failure

ROCKI

NG:

4

State-of-the-art: Unreinforced masonry walls

• Determining the displacement capacity

from quasi-static cyclic tests

V

D

• Assemble data bases + fit empirical

models

5

State-of-the-art: Unreinforced masonry walls

Empirical model (EC8)

Walls failing in flexure Walls failing in shear

Masonry typology:

Clay brick masonry with

normal cement mortar

• Poor prediction of mean response

• Very large scatter

Prediction of drift capacity with empirical model d=D/H

1.8±1.1 1.7±1.0

6

Drift capacity of masonry walls

• Biased towards unrepresentative small test units

• General problem of structural engineering: Large tests required

number of tests will always be limited

Limits for empirical models

Drift capacity

reduces with the

size of the test

unit.

7

Change the scale at which we measure and predict

Global response

Local response

PUP3: g at Vmax

V D

8

Analytical model of masonry walls: Critical Diagonal Crack Model

9

0 0

50

100

150

200

PUP1 Model

Test pos. env.

0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 0

0.2

0.4

0.6

0.8

1

u f l /u Model

u s h

/u Model

u f l /u Test

u s h

/u Test

Drift δ [%]

Sh

ea

r fo

rce V

[kN

]

Horiz. displ. u [mm]

Dis

pla

ce

ment co

mp

one

nts

[-]

PUP1

Validation of analytical model

against experimental results on the

global and local level

Analytical model of masonry walls: Critical Diagonal Crack Model

10

Empirical model (EC8)

Analytical model

Walls failing in flexure Walls failing in shear

Performance of empirical vs analytical model

Drift capacity of

masonry walls

Research results

Proof of concept that developing such analytical models for predicting the

drift capacity of unreinforced masonry walls is feasible.

Better because they connect local to global deformation measures

1.8±1.1 1.7±1.0 0.7±0.3 0.9±0.4

11

Analytical model of masonry walls: Critical Diagonal Crack Model

12

Suite of cycles with increasing amplitudes

• Variables:

• Number of cycles per drift amplitude (often n=2 or n=3)

• Increase of amplitudes Dd

n=3 Dd

Loading history

13

Does the choice of the loading history influence the obtained drift

capacities?

Systematic study on cumulative damage effects on URM walls missing

Pairs of monotonic and cyclic tests:

Ganz & Thürlimann (1984)

• W1 (monotonic) & W6 (cyclic)

• W2 (monotonic) & W7 (cyclic)

Magenes & Calvi (1992)

• MI1m (monotonic) & MI1 (cyclic)

Loading history

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• Force capacity is not affected by cumulative damage demand

• Displacement capacity from monotonic tests is twice as large as from

cyclic tests.

Displacement capacity depends on the demand.

Does the choice of the loading history influence the obtained drift

capacities?

15

• Flexural failure: Unaffected by number of cycles

• Shear and hybrid failures: Displacement capacity reduces with increasing number of cycles that are applied

Displacement history that is applied in cyclic tests matters.

Loading history: Influence of number of cycles on du

16

• Many protocols have been proposed:

• One specific for URM structures: SPD protocol (Porter 1987),

result of US-Japan workshop

• One that is most frequently used: SAC protocol (ATC 1992)

Loading proctocols for quasi-static cyclic tests

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• All existing loading protocols derived for regions of high seismicity

• When applied to structural elements for regions of moderate

seismicity: Test units subjected to too many cycles

Sion Basel

Distance [km]

Ma

gn

itu

de

Distance [km]

Mag

nit

ud

e

Existing protocols

De-aggregation Results for Switzerland for T=2475 yrs (2/50)

Sion (agd=0.15g for T=475 yrs) Basel (agd=0.12g for T=475 yrs)

Sion Basel

Loading proctocols for quasi-static cyclic tests

18

New loading protocols: Moderate vs. High Seismicity

2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Number of cycles

Norm

aliz

ed c

ycle

am

plit

ude

f(t)=0.50-0.55exp(t2.4

)

f(t)=-0.50+0.55exp(t2.4

)

T=0.2s

2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Number of cycles

No

rma

lize

d c

ycle

am

plit

ud

e

f(t)=-0.50+0.55exp(t2,2

)

f(t)=0.50-0.55exp(t2,2

)

T=0.3s

Mo

dera

te

seis

mic

ity

Hig

h

seis

mic

ity

5 10 15 20 25-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Number of cycles

No

rma

lize

d c

ycle

am

plit

ud

e

5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Number of cycles

Norm

aliz

ed c

ycle

am

plit

ude

ntot=14

α=2.4

ntot=10

α=2.2

ntot=24

α=2.8

ntot=20

α=1.9

Loading proctocols for quasi-static cyclic tests

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Summary & Outlook

Summary

• Analytical drift capacity models seem feasible (for all limit

states)

• As long as we do not have a model that captures cumulative

damage effects on masonry performance, we should test with

realistic testing protocols.

Future research

• Effect of cyclic degradation / cumulative damage

• Effect of strain rates

• Failure criteria for other masonry typologies (thin bed mortar,

stone masonry, …)

eyy

eyy

gxy

gxy

20

Change the scale at which we measure and predict

Coupling of shear and flexural deformations

Axial strain profiles

Shear strains

concentrate in

the compression

strut

Linear strain

profile in

compression

zone is a

reasonable

assumption

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Criteria for onset of crushing:

Confinement effect at the base of the wall

Onset of crushing in 2nd joint

Strength of masonry fu

determinant in 2nd joint

Strength of brick fcB

in base joint

Change the scale at which we measure and predict

22

Shear critical walls: Critical Diagonal Crack Model

Assumptions:

- Deformations in diagonal crack can be lumped in a single diagonal crack

(“Critical Diagonal Crack” = CDC)

- Geometry of CDC:

- Geometry of wall

- Geometry of bricks V

N

M

Critical diagonal crack

Analytical model of URM walls: CDC Model

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Onset of formation of CDC: Modified Mann – Müller criterion

- Stress demand: Plane section analysis

Compute position along the CDC where the vertical bed joint stresses

due to N, M, T are for the first time zero

Location where diagonal crack starts to form

τxy

σN

σT

L

H0

H-H0

H

N

M(x=H)

V

x

y

τxy

(y,x=Hcrit )

σN

σT

lB

hB

σM

σM

H crit

τxy

τxy

σN

σT

L

H0

H-H0

H

N

M(x=H)

V

x

y

τxy

(y,x=Hcrit )

σN

σT

lB

hB

σM

σM

H crit

τxy

Analytical model of URM walls: CDC Model

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Analytical model of URM walls: CDC Model

Peak strength: Mohr-Coulomb criterion based on local stresses

- Stress demand:

- Axial stresses: Plane section analysis

- Shear stresses: Parabolic stress distribution, shear only carried by area in

compression

- Peak strength reached when

- Mohr-Coulomb criterion is exceeded

- Compressive strength is exceeded (confinement effect considered)

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Analytical model of URM walls: CDC Model

Stiffness of wall

• Only area in compression contributes to the stiffness

• The diagonal crack divides the wall section into two sections

(composite beam)

• Limited shear stress transfer between the sections allowed

y

τxy

(y)σxx

(y)

y

L1 L2

L

y y

Lc Lc

L1 L2

L

σxx

(y) τxy

(y)

L

H

x

L2(x)L1(x)

y

Uncracked

section

Flexural

decompression

Shear cracking

g-factor approach

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Analytical model of URM walls: CDC Model

Residual strength: Zero stress transfer in CDC

- Stress demand: - Axial stresses: Plane section analysis

- Shear stresses: Parabolic stress distribution, shear only carried by area in compression

- Residual strength reached when - Tensile strength of top and bottom brick exceeded (Turnsek-Cacovic)

- Compressive strength is exceeded (confinement effect considered)