lesson9 2nd part

Masonry Structures, lesson 9 part 2 slide 1

Seismic design and assessment ofMasonry Structures

Seismic design and assessment ofMasonry Structures

Lesson 9, continuedOctober 2004


To perform a separate analysis for each storey, it is necessary to make assumptions on the boundary conditions of the piers, i.e. on their rotational restraints: fixed-fixed, or fixed-free, or other.

These assumptions are strongly affected by the strength and stiffness of the coupling horizontal structural elements: plain unreinforced masonry spandrel beams, or r.c. slabs, or r.c. ring beams, which may or may not crack or fail as horizontal loads increase.

The state of stress of these elements cannot be determined accurately on the basis of a separate analysis for each storey, but only from a global analysis of the whole multi-storey structure. In principle, only by knowing how much the coupling element are stressed can the engineer judge if cracking or failure can be expected, and, as a consequence, what kind of boundary conditions can be assumed for the piers.

A variation in the axial force of the piers may take place under the overturning effect of the horizontal loads, affecting the flexural and shear strength of the individual piers. This effect may not be of relevance in low-rise squat buildings, but it can be in a more general context. Again, an evaluation of this effect can be made only very approximately with a separate storey-by-storey analysis.

Limitations of the storey mechanism approach


The storey-mechanism approach must therefore always be applied with a clearunderstanding of its meaning and limitations, otherwise it can lead, in some cases, to unrealistic and unconservative results.

The engineer can improve to some extent the results with a proper choice of boundary conditions (end rotation) for the piers, but still some structural configurations of multi-storey walls or buildings cannot be analysed properly with such method.

Limitations of the storey mechanism approach


URM MASONRY SPANDREL BEAMS UNDER SEISMIC ACTION

Crack patterns from an experimental cyclic test on a full-scale masonry building prototype (University of Pavia, 1994)

at first cracking

at ultimate


Very little information is available on the behaviour of urm spandrel beams subjected to cyclic shear. A proposal for strength evaluation which could be suitable for applications is as follows.

Unreinforced masonry spandrels can be considered as structurally effective only if they are regularly bonded to the adjoining walls and resting on a floor tie beam or on an effective lintel.

The verification of unreinforced masonry coupling beams, in presence of a known axial horizontal force, is carried out in analogy of the vertical walls.

If the axial load is not known from the model (for instance, when the analysis is carried out with the hypothesis of in-plane infinitely rigid floors), but horizontal elements with tensile strength (such as steel ties or r.c. ring beams) are present in proximity of the masonry beam, the resisting values may be assumed not greater than the following values associated to the shear and flexural failure mechanisms.

Strength of urm spandrel beams


The shear strength Vt of an unreinforced masonry coupling beam, connected to a r.c. ring beam or a lintel and effectively bonded at the ends, may be computed in a simplified way as follows:

Vt = h t fv0

where: h is the section height of the masonry beam;

t is the width (thickness) of the beam

fv0 = is the shear strength in absence of compression.



The maximum resisting moment, associated to the flexural mechanism, always in presence of horizontal elements resisting to tension actions in order to balance the horizontal compression in masonry beams, may be evaluated as follows:

where: Hp is the minimum between the tension strength of the element in tension placed horizontally and the value 0.4fhuht

fhu= is the compression strength of masonry in the horizontal direction (in the plane of the wall).

The shear strength, associated to this mechanism, may be computed as:

where l is the clear span of the masonry beam.

The value of shear strength for the unreinforced masonry beam element shall be assumed as the minimum between Vt and Vp.


[ ])85.0/(12/ htfHhHM huppu −=

lMV up /2=


Non linear static modelling: beyond the storey mechanism approach

“Storey mechanism”

Tomaževič, Braga & Dolce

Ok up to 2 (3?) storeys

Refined finite element

Gambarotta & Lagomarsino, Papa & Nappi., Lourenço,…Macro-element modelling

SAM (Magenes, Della Fontana, Bolognini)

PEFV (D’Asdia & Viskovic)

MAS3D (Braga,Liberatore, Spera)

TREMURI (Lagomarsino, Penna & Galasco)

fascia

maschio

nodo


Requirements for non linear models

• Low or moderate computational burden to allow the modeling of whole buildings:• discretization of the structure with macro-elements: the elements have

dimensions comparable to the inter-storey height or with the size of openings (doors, windows), to reduce the number of degrees of freedom of the model.

• Reliability of results:

• all the fundamental failure mechanisms should be accounted for with suitable failure criteria;

• the model should give a good estimate of the overall deformational behaviour under horizontal loads.


Overview of some macroelement models for urm

EQUIVALENT TRUSS APPROACH (Pagano et al., 1984-1990)



MULTI-FAN MODEL, MAS3D (Braga, Liberatore, Spera, 1990-2000)No-tension stress field simulated as a set of “radial” stress fields for which an analytical formulation in closed form exists.



PEFV (D’Asdia & Viskovic 1990-today)Linear elastic finite elements with variable(adaptive) geometry.

Pier or spandrel elem.

“Joint” element



TREMURI(Lagomarsino, Penna, Galasco 1997- today)

Beam-columns-type elements with internal degrees of freedom and coupling of rotation/axial displacement to simulate rocking. Allows dynamic analysis also.


Overview of some macroelement models for urmSAM (Magenes, Della Fontana, Bolognini 1998- today)

Equivalent 3–d frame model

•Simplified strength criteria for all elements, including r.c. ring beams, easily adaptable to code-like formulations.

•Simplified multi-linear constitutive rules are used (extension of concepts already present in early storey-mechanism formulations)

•Flexural (“rocking”) failure:a plastic hinge is introduced at the end of the effective length where Mu is attained

•Shear failure: plastic shear deformation γ occurs when Vu is attained

•Suitable for both urm and reinforced masonry.

•Crude idealization but effective results especially for prediction of behaviour at ultimate


Nonlinear equivalent frame

γ

V

Vu

γ = θ − ϕ u

H1

Heff

H2

rigid offset

effectivelength

i

i '

j

j 'rigid offset

θ = chord rotation

ϕ = flexural deform.

γ = shear deformation

Shear force-shear deformation behaviour in the case of shear failure mechanism γ

V

Vu

γ γ 1 2

α Vu

Pier elementSpandrel element


0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

Total displacement at 3rd floor (m)

0

10

20

30

40

50

60

70

80

Tota

l bas

e sh

ear (

kN)

F.E.M. SAM (w. brittle spandrels)


Damage pattern predicted by refined nonlinear f.e.m. analysis

URM wall with weak spandrels: No storey mechanism



0 5 10 15 20 25Equivalent displacement δeq (mm)

0

20

40

60

80

100

120

140

160

Bas

e sh

ear

(kN

)

Exp. 1st cycle envelopeExp. 2nd cycle envelopeExp. 3rd cycle envelopeSAM pushover analysis

-25 -20 -15 -10 -5 0 5 10 15 20 25

Equivalent displacement δeq (mm)

-150

-100

-50

0

50

100

150

Bas

e sh

ear (

kN)

Wall D - Door wall

Comparison with experiments: full scale, two-storey, brick masonry building, subjected to quasi static cyclic loading (University of Pavia, 1994-95)


29.26

2.03

19.12 2.25

2.25

1.45

2.25

1.63

0.64

1.052.731.051.741.053.70

1.22

2.25

1.45

2.25

1.45

3.701.051.741.052.731.052.032.56

5-storey urm wall with r.c. ring beams


5-storey urm wall with r.c. ring beams: equivalent frame model


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Roof displacement (m)

0

200

400

600

800

1000

1200

1400

Tota

l bas

e sh

ear (

kN)

Analysis A

Analysis G

Analysis C

Analysis B

0.00

0.06

0.12

0.18

0.24

0.30

0.36

0.42

Bas

e sh

ear c

oeff

icie

nt

0.000 0.078 0.156 0.234 0.312 0.390 0.468

Global angular deformation (%)

Pushover analysis withfirst-mode (linear) force distribution.

R.c. beams: elasto-plastic beam elements (w. flexural hinging).

The analyses from A to Gshow the effect of decreasing strength and stiffness of the r.c. beamson the response of the wall.

No r.c. ring beams

5-storey urm wall: nonliner equivalent frame pushover analysis


5-storey urm wall: nonliner equivalent frame pushover analysis

Coupling elements (masonry spandrels and r.c. beams) can affect not only the strength, but also the overall deformed shape and collapse mechanism

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Horizontal displacement (m)

0

4

8

12

16

20

Hei

ght (

m)

Analysis AAnalysis CAnalysis G

1st FLOOR

2nd FLOOR

3rd FLOOR

4th FLOOR

5th FLOOR

soft storey

soft storey

global overturningof cantilever walls



Forza alla base-Spostamento

0

100

200

300

400

500

600

700

800

900

1000

0 0.01 0.02 0.03

Spostamento [m]

Forz

a [K

N]

POR

SAM

Comparison of a 3-d storey –mechanism analysis and a 3-d nonlinear frame analysis: two storey urm building with rigidfloor diaphragms and r.c. ring beams.

The flexural and shear strength criteria of masonry walls are kept the same for both methods

SAM

storey mechanism


Use of nonlinear static analysis in seismic design/assessment

The non linear static analysis is based on the application of gravity loads and of a horizontal force system that, keeping constant the relative ratio between the acting horizontal forces, is scaled in order to monotonically increase the horizontal displacement of a control point on the structure (for example, the centre of the mass of the roof), up to the achievement of the ultimate conditions.

A suitable distribution of lateral loads should be applied to the building. At least two different distributions must be applied:

-a “modal” pattern, based on lateral forces that are proportional to mass multiplied by the displacement associated to the first mode shape- a “uniform” pattern, based on lateral forces that are proportional to mass regardless of elevation (uniform response acceleration).

Lateral loads shall be applied at the location of the masses in the model, taking into account accidental eccentricity.



The relation between base shear force and the control displacement (the “capacity curve”) should be determined by pushover analysis for values of the control displacement ranging between zero and a sufficiently large value, which must exceed by a suitable margin the displacement demand which will be estimated under the design earthquake (target displacement) .

The target displacement is calculated as the seismic demand derived from the design response spectrum by converting the capacity curve into an idealized force-displacement curve of an equivalent single-degree-of-freedom system.

For the evaluation of the displacement demand of the equivalent s.d.o.f. system, different procedures can be followed, depending on:• how the seismic input is represented (acceleration spectra, displacement spectra, composite A-D spectra);• how the inelastic and hysteretic behaviour of the structure is accounted for (equivalent viscous damping, ductility demand, energy dissipation demand).



An example of procedure (e.g. as adopted by EC8 and Italian code):


0

100

200

300

400

500

600

700

800

0 0.01 0.02

Spostamento [m]

Forz

a [K

N]

T ET T O705

564

0.0146

ULSDLS

Roof displacement (m)

Bas

e sh

ear

(kN

)

Step 1: carry out the pushover analysis with the chosen force distribution. Plot capacity curve and determine the performance limit states of interest



0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30Roof displacement [cm]

Bas

e sh

ear[

kN]

dc

Fb

∑∑=Γ

Φ

Φ2ii

ii

m

m

Γ= bFF*

Γ= cdd*

**2*

kmT π=∑

=

Φ=N

iiimm

1*

array that represents the mass displacement in the first mode of vibration of the structure, in the considered direction, normalized to the unit value of the relative component of the control point.

Φ

Step 2: determine an equivalent bilinear s.d.o.f. system




0

100

200

300

400

500

600

700

800

900

0 0.01 0.02Spostamento [m]

Forz

a [K

N]

Sistema equivalente SDOFTETTOBilineare

F*max

F*y

d*max

0.7F*max

0.8F*max

d*y

Bas

e sh

ear

(kN

)

Displacement (m)

Capacity curve

Equivalent bilinear SDOF



Γ= cdd*

**2*

kmT π=∑

=

Φ=N

iiimm

1*

*)(max,max TSdd Dee ==∗if T*≥TC

( ) max,max,

max *1*1

* eCe d

TTq

qd

d ≥⎥⎦⎤

⎢⎣⎡ −+=∗

if T*<TC

*

*** )(

y

e

FTSmq =

∗maxd

Elastic displacement spectrum

elastic accelerationspectrum

Step 3: using the elastic response spectrum, calculate the displacement demand on the sdof system



Step 4: convert the displacement demand on the equivalent sdof into the control displacement and find target point on capacity curve and compare with displacement capacity.

max,*max cdd =Γ

∗maxd

Stato Limite DS

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30

Spostamento copertura [cm]

Tagl

io a

lla b

ase

[kN

]max,cd



Available on ftp site:

Relevant chapters of new Italian seismic code

(English translation available! Thanks Paolo)

Relevant chapters of FEMA 356

Eurocode 8 (see Annex B)


When and how to use storey-mechanism method

Eurocode 8: “For low-rise masonry buildings, in which structural wall behaviour is dominated by shear, each storey may be analyzed independently. Such requirements are deemed to be satisfied if the number of storey is 3 or less and if the average aspect ratio of structural walls is less than 1.0.

….

New Italian seismic code: “For buildings with number of storeys greater than two, the structural model should take into account the effects due to the variation of the vertical forces due to the seismic action and should guarantee the local and global equilibrium. “


Earlier use of storey-mechanism method (Tomaževič)

212 +

≥q

uµ

qS

gaWkqTSWVH totjdtotjjdesignjdu

0,, );( βυ ⋅

⋅⋅=⋅⋅=≥

e

u

e

uu d

dΦΦ

==µ

q behaviour factor (force reduction factor), specified by code (e.g. 1.5-2.0 for urm)

Ultimate ductility

Φ = d/h storey drift


Use of storey-mechanism method with present EC8 procedure

Fel,base = Se(T1) Wtot /g = Se(T1) Mtot

•Evaluate elastic period of building T1 , e.g. using approximate formulae.

•Estimate elastic base shear from elastic acceleration spectrum:

•Evaluate ratio between interstorey shear Vj of the storey j being considered and the total base shear:

where Fi is the seismic force at the i-th floor.

∑=

=N

jiij FVbasejj FV /=υ

•The equivalent sdof is defined by putting F* = Vj and d*= interstorey displacement

•Evaluate q* = υj Fel,base /F*y

•Calculate d*max= d*y [1+(q*-1)Tc/T1] (not greater than q d*y ) and check d*max≤ du

lesson9 2nd part

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