Dan Abrams + Magenes Course on Masonry

1. Overview of some macroelement models for urm
SAM (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
Masonry Structures, lesson 9 part 2 slide 4
Nonlinear equivalent frame
rigid i
offset
H1 θ = chord rotation
i'
ϕ = flexural deform.
effective
γ = shear deformation
length Heff
j'
rigid H2
offset
j
V
V
Spandrel
Shear force- Pier element element
V
shear V u
u
deformation
behaviour in
the case of αV
u
shear failure
mechanism γ
γ γ γ
1 2
γ = θu− ϕ
Masonry Structures, lesson 9 part 2 slide 5

2. Nonlinear equivalent frame
Comparison of a 3-d storey – 1000
Forza alla base-Spostamento
mechanism analysis and a 3-d 900
storey mechanism
nonlinear frame analysis: two POR
SAM
800
storey urm building with rigid
700
floor diaphragms and r.c. ring
600
beams. SAM
Forza [KN]
500
The flexural and shear strength 400
criteria of masonry walls are 300
kept the same for both methods 200
100
0
0 0.01 0.02 0.03
Spostame nto [m]
Masonry Structures, lesson 9 part 2 slide 6
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.
Masonry Structures, lesson 9 part 2 slide 7

3. Use of nonlinear static analysis in seismic design/assessment
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).
Masonry Structures, lesson 9 part 2 slide 8
Use of nonlinear static analysis in seismic design/assessment
An example of procedure (e.g. as adopted by EC8 and Italian code):
Forza alla base-Spostamento
800
T ET T O
705
Step 1: carry out 700
the pushover 600 DLS
analysis with the 564 ULS
Base shear (kN)
chosen force 500
Forza [KN]
distribution. Plot 400
capacity curve
300
and determine the
performance 200
limit states of
100
interest
0
0 0.01 0.0146 0.02
Spostamento [m]
Roof displacement (m)
Masonry Structures, lesson 9 part 2 slide 9

4. Use of nonlinear static analysis in seismic design/assessment
Γ = ∑m iΦ 2i
Φvibration of the structure,mass displacementdirection, normalized
array that represents the
in the considered
in the first mode of
mΦ
to the unit value of the relative component of the control point.
∑ i i
2000 Fb
Fb
1800
F* =
1600 Γ
Base shear [kN]
1400
1200 Step 2: determine an
1000 equivalent bilinear
800
600
s.d.o.f. system
400
200 dc
0
dc
0 5 10 15 20 25 30 d* =
Roof displacement [cm] Γ
N m*
m* = ∑ mi Φ i T * = 2π
i =1 k*
Masonry Structures, lesson 9 part 2 slide 10
Use of nonlinear static analysis in seismic design/assessment
Forza alla base-Spostamento
900
800
Capacity
F*max curve
700
F*y
600
0.8F*max
0.7F*max
500 Sistema equivalente SDOF
TETTO
Forza [KN]
Equivalent
Base shear (kN)
400 Bilineare
bilinear
SDOF
300
200
100
0
0 d*y 0.01 d*max 0.02
Displacement
Spostamento [m] (m)
Masonry Structures, lesson 9 part 2 slide 11

5. Use of nonlinear static analysis in seismic design/assessment
Elastic displacement spectrum
Step 3: using the elastic
response spectrum, calculate
∗
the displacement demand on if T*≥TC d max = d e , max = S De ( T *)
the sdof system
if T*<TC
d e , max ⎡ T ⎤
∗
d max = 1 + (q * − 1 ) C ⎥ ≥ d e , max
q* ⎢ ⎣ T *⎦
elastic acceleration
m* S e (T * ) spectrum
q =
*
d* =
dc Fy*
∗
d max Γ
N m*
m* = ∑ mi Φ i T * = 2π
i =1
k*
Masonry Structures, lesson 9 part 2 slide 12
Use of nonlinear static analysis in seismic design/assessment
Step 4: convert the displacement demand on the
equivalent sdof into the control displacement
and find target point on capacity curve and Γd max = d c ,max
*
compare with displacement capacity.
2000
Stato Limite DS
1800
1600
Taglio alla base [kN]
1400
1200
1000
800
600
400
∗ 200
d max d c , max
0
0 5 10 15 20 25 30
Spostamento copertura [cm]
Masonry Structures, lesson 9 part 2 slide 13

6. Modeling and Acceptability Criteria
Nonlinear Static Procedure
e
d primary walls
force
LS CP
secondary walls
0.75d
LS CP c
0.75e
drift
Masonry Structures, lesson 9 part 2 slide 14
NSP: Acceptable Drifts
Drifts for Primary Elements
IO LS CP
Bed-Joint Sliding 0.1% 0.3% 0.4%
Rocking 0.1% 0.3 heff/L% 0.4 heff/L%
(Multiply drifts by 2 for secondary elements for LS and CP)
Masonry Structures, lesson 9 part 2 slide 15

7. FEMA 356: evaluation of target displacement, NSP
Te2
δ t = C0C1C2C3 S a 2 g
4π elastic
displacement
response spectrum,
Te evaluated with
stiffness “at yield”
to relate roof modification
modification modification
displacement to factor to account
factor to factor to account
sdof estimate for dynamic P-∆
displacement for different effects (1.0 for
inelastic hysteretic
same as Γ displacement stiff structures)
behaviour
same formulae not mentioned in
not mentioned in EC8 procedure
as with q* EC8 procedure
Masonry Structures, lesson 9 part 2 slide 16