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Nonlinear seismic analysis of masonry buildings · PDF file 2013-10-01 · for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799) Pushover

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  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Nonlinear seismic analysis of masonry buildings

    Erlenbach, September 12th, 2013

    Department of Civil Engineering and Architecture

    University of Pavia, Italy

    Andrea Penna [email protected]

    EUCENTRE  Foundation

    • Highly nonlinear behaviour

    • Need for nonlinear analysis recognized since late 1970s (Tomazevic, 1978; Braga and Dolce, 1982)

    • Pushover analysis

    • Equivalent frame modelling

    Seismic response of masonry buildings

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Global seismic analysis of masonry buildings • Modelling of the mechanical behaviour • (Nonlinear static) pushover analysis • Models for pushover analysis • Mixed masonry-r.c. buildings

    Modelling of the mechanical behaviour

    T

    N

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Modelling of the mechanical behaviour

    Flexure-rocking

    Shear-sliding (friction)

    Seismc response

    -100

    -80

    -60

    -40

    -20

    0

    20

    40

    60

    80

    100

    -8 -6 -4 -2 0 2 4 6 8

    displacement (mm)

    fo rc

    e (k

    N )

    Cyclic behaviour: stiffness degradation and strength

    deterioration

    0.4 0.8-0.4-0.8

    40

    20

    -20

    -40

    0

    First-Story Drift, %

    St or

    y Sh

    ea r,

    kN

    0

    Dynamic response

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Analysis of the seismic response

    • Earthquake-resistant structure: walls + floor diaphragms

    • Walls  resisting elements (both vertical and horizontal loads)

    • Floor diaphragms share vertical loads on walls and are in-plane stiffening elements

    • Out-of-plane behavior of walls and flexural response of floors negligible with respect to the global behavior (under certain conditions)

    • Highly nonlinear behaviour

    • Computational approaches

    Pushover analysis

    • Seismic demand (seismic action) • Structural capacity (capacity curve) • Performance  Displacement limit states • Definition of an equivalent nonlinear SDOF system • Choice of the horizontal loading pattern • Global assessment

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Pushover analysis

    Representation of seismic action

    • Acceleration and displacement response spectra • Spectral coordinates • Seismic response of nonlinear systems • Inelastic spectra • Reduction factors and ductility demand

    Pushover analysis

    Acceleration and displacement response spectra

    Se

    TTb Tc Td

    SD

    TTb Tc Td

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Pushover analysis

    Spectral coordinates SA

    TTb Tc Td SD

    TTb Tc Td

    SD

    Tb Tc

    Td

    SA

    2

    2 2 )( )(

      

      

    TTS TS

    D

    A 

    Pushover analysis

    Seismic response of nonlinear systems

    F

    Ddy dmax = dy

    “Rigid” structures

    Fe

    Fy

    F

    Ddy dmax = dy

    “Flexible” structures

    Fe

    Fy

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Pushover analysis

    Ductility demand and spectral reduction factors

     

     

    

      

       

     

    c y

    e R

    c c

    y

    e R

    TTse F F

    TTse T T

    F F

     11 (Fajfar, 1999)

      

      

    

    

    cD

    c c

    D

    TTseq

    TTse T Tq

     11

    y

    e

    y

    e

    F TmS

    F Fq )(

    Spectral reduction coefficient or “behaviour

    factor”

    Pushover analysis

    Displacement demand for a «rigid» system

    SD

    SA

      max,max,max 11 eCe dT Tq

    q d

    d  

      

    m Fy

    dmax de,max

    y

    A

    y

    e

    F TmS

    F Fq )(

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Pushover analysis

    Structural capacity

    • Base shear – reference displacement • Capacity curve • Spectral coordinates

    PUSHOVER ANALYSIS

    • Basic idea of the method: apply an horizontal force distribution to the structural model to directly evaluate its nonlinear (static) response

    • Hypothesis: the lateral response of the structure under the effect of a properly incremented vector of horizontal forces can be assumed as the envelope of the possible response obtained by nonlinear time-history analysis

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    PUSHOVER ANALYSIS

    Base shear

    F1

    Fi

    Fi+1

    Fn

     n

    ib FT 1

    dtop

    PUSHOVER ANALYSIS

    Capacity curve

    DTOP

    TB

    SA SD

    SA

    SD

    SA

    SD

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Pushover analysis

    Performance Displacement limit states

    • Performance limit states • Damage limit states for structural members • Interstorey drift ratio • Damage limitation • Ultimate limit states

    Pushover analysis

    Analysis results:

    • Capacity curve • Limit states: from local element damage to global limit states • Safety assessment in terms of global displacements

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    Simple model for masonry structural members

    Du = 0.004-0.008 hDy

    VR

    VR

    N

    22

    FLEXURAL STRENGTH

    In-plane bending failure ↔ toe-crushing

    For relatively low compression values (N) the wall tends to overturn similarly to a rigid body

    The analysis of the wall bending response can be based on an appropriate definition of a “stress-block” for the compressed part of the masonry cross section

    tf Na u

     

      

     

      

     

     

       

    u

    mm

    u u f

    tl ltf

    NNlalNM  

     1

    2 1

    22

    2

    Vertical translation:

    Rotation :  = 0.85-1

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    23

    Flexural strength

    esup

    einf

    P V

    V P

    H H

    0

    D

    e P

    a

    D/2D/2

    x

    M=Pe=VH0

    fu

    Dt Pp

    f pDPMePHV

    u u 

      

     

      ;

    1

    2inf0max 

    24

    Cyclic shear response

    -100

    -80

    -60

    -40

    -20

    0

    20

    40

    60

    80

    100

    -8 -6 -4 -2 0 2 4 6 8

    displacement (mm)

    fo rc

    e (k

    N )

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    25

    SHEAR STRENGTH

    The definition of “shear failure” usually includes different cracking modes associated with the combined effect of shear and compression stress

    Two main shear failure modes can be identified:

    a) diagonal-cracking

    b) shear-sliding

    Diagonal crackig: weak joints

    Diagonal cracking: strong joints

    26

    Shear strength (1)

    Dt Pp

    f p

    b DtfV

    tu

    tu u

    

    ; 1 P

    V

    ftu = tensile strength

    (Turnsek & Sheppard, 1980)

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    27

    Shear strength (2)

     

      

     

     

      

     

     

      

     

    pc pcDtpcDt

    Dt PcDtV

    V

    u

     

     

     

    31 5.1

    Strength of the cracked section:

    Sliding on bed-joints:

      c P

    V

    28

    Shear-compression interaction diagram

    0 100 200 300 400 500 600 700 800 900

    1000 1100 1200

    0 10 20 30 40 50 60 70 80 90 100

    N/Nu [%]

    V re

    s [ kN

    ]

  • A. Penna – Software Forum - Erlenbach Sept. 12, 2013

    (Lagomarsino S, Penna A, Galasco A, Cattari S [2013] TREMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799)

    Pushover analysis

    Analysis control A pushover analysis consists of applying to the structure gravity loads and a system of of distributed horizontal forces in the considered analysis direction, at each building level, proportionally to the inertial masses (sum of the horizontal forces = base shear).

    Such forces are scaled t

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