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STATIC NONLINEAR
ANALYSIS
Advanced Earthquake Engineering
CIVIL-706
Instructor:
Lorenzo DIANA, PhD
1
By the end of today’s course…
Static nonlinear analysis Advanced Earthquake Engineering CIVIL-706
You will be able to answer:
• What are NSA advantages over other structural analysis methods?
• How to perform an NSA?
• What are the key elements when performing NSA?
2
Advanced Earthquake Engineering CIVIL-706
Earthquake Engineering Assessment
Static nonlinear analysis
Sou
rce:
FEM
A 4
40
3
Advanced Earthquake Engineering CIVIL-706
Earthquake Engineering Assessment
Action Structure
Static Dynamic
Linear Linear static
analysis Linear dynamic
analysis
Non-linear Non linear static
analysis Non linear
dynamic analysis
Static nonlinear analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
4
Advanced Earthquake Engineering CIVIL-706
Linear static analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis
• Seismic forces are applied as equivalent static forces
• Regular (plan and vertical) buildings
• NTC 2008: T1=C1 x H3/4
T1 period of the structure
C1 factor related to material
H height of the structure
5
Advanced Earthquake Engineering CIVIL-706
Equivalent Force Method
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis
Fi = Fh x zi x Wi / ∑ zj x Wj
Fh = Sd (T1) x W x λ / g
W total mass of the structure
λ factor related to the number of storyes
Wi and Wj mass of i and j
zi and zj height of i and j
6
Advanced Earthquake Engineering CIVIL-706
Equivalent Force Method
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis 7
Advanced Earthquake Engineering CIVIL-706
Linear dynamic analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis
• Seismic forces are applied as a dynamic action on structures
• Regular (plan and vertical) buildings
• NO mix structures
• M · ẍ + C · ẋ + K · x = - M · ex · ẍg
M mass matrix
C damping matrix
K stiffness matrix
ex vector of seismic direction
8
Advanced Earthquake Engineering CIVIL-706
Linear dynamic analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis
• Response spectrum method
• Overlapping of effects (SRSS: square root of the sum of the square)
9
Advanced Earthquake Engineering CIVIL-706
Linear dynamic analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis 10
Advanced Earthquake Engineering CIVIL-706
Linear dynamic analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis 11
Advanced Earthquake Engineering CIVIL-706
Linear dynamic analysis
Action Structure
Static Dynamic
Linear Equivalent Force
method Response
Spectrum method
Non-linear Pushover Non-linear Dynamic
Static nonlinear analysis 12
Advanced Earthquake Engineering CIVIL-706
Non linear time history analysis
• Advantage: the complexity of the dynamic load is considered + the dynamic behavior of structure
• Disadvantage: time consuming
Static nonlinear analysis 13
Advanced Earthquake Engineering CIVIL-706
Non linear static procedure
• How does it work? Capacity vs. Demand
Static nonlinear analysis
Sa
Sd
14
• Necessary elements?
– Acceleration-displacement response spectrum (ADRS)
– Capacity Curve
Advanced Earthquake Engineering CIVIL-706
Nonlinear static procedure
Static nonlinear analysis
V (force)
Δ (disp.)
Sa
Sd
15
Advanced Earthquake Engineering CIVIL-706
Nonlinear static procedure
• Developing capacity curves:
– Displacement-based methods
Static nonlinear analysis 16
Advanced Earthquake Engineering CIVIL-706
Nonlinear static procedure
• Assumptions
– Response: maximum displacement
– Deformation: “most often” following the first mode
• Benefits
– Displacement-based analysis
– Nonlinear behavior of the structure
• Drawbacks
– Simplified approach (static)
– Damping difficult to represent
Static nonlinear analysis 17
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 18
Advanced Earthquake Engineering CIVIL-706
Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 19
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
• Capacity curve defines the capacity of the structure independent to any seismic demand
• Objectives: – Estimate the maximum horizontal displacement
– Estimate the (global) ductility of the structure
Static nonlinear analysis
Structural model (FEM, AEM, etc.)
This model allows to define, through few geometrical and mechanical parameters, a capacity curve, which is representative of the structure response in non-linear field and to obtain a simplified assessment of the structure overall strength considering only a walls in-plane behavior and taking into account two different collapse modes known as uniform and soft story.
20
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
• Pushover
• Simplified analytical method (EC8)
• DBV method
Static nonlinear analysis
This model allows to define, through few geometrical and mechanical parameters, a capacity curve, which is representative of the structure response in non-linear field and to obtain a simplified assessment of the structure overall strength considering only a walls in-plane behavior and taking into account two different collapse modes known as uniform and soft story.
L’analisi di pushover o analisi di spinta (letteralmente pushover significa “spingere oltre”) è una procedura statica non lineare impiegata per determinare il comportamento di una struttura a fronte di una determinata azione (forza o spostamento) applicata. Essa consiste nello “spingere” la struttura fino a che questa collassa o un parametro di controllo di deformazione non raggiunge un valore limite prefissato; la “spinta” si ottiene applicando in modo incrementale monotono un profilo di forze o di spostamenti prestabilito.
21
• Pushover
Load pattern: distribution of forces should represent the dynamic behavior
– Triangular (if building regular enough) – Following the first mode – Other simplified distributions depending on the building
Base Shear
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
Static nonlinear analysis 22
• Pushover
Load pattern: importance definition
– Effect of lateral load can change the capacity curve
Base Shear
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
• Pushover
Load pattern: importance definition
– Effect of lateral load pattern
Static nonlinear analysis 23
• Pushover
Different model type
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
• Pushover
Load pattern: importance definition
– Effect of lateral load pattern
Static nonlinear analysis
Structural model (FEM, AEM, etc.)
24
Advanced Earthquake Engineering CIVIL-706
Capacity Curve
• Force/Displacement to spectral coordinates
Static nonlinear analysis
V
Δ
Sa
Sd
1
1 ,
1
(m o d .1)
(m o d .1)
(m o d .1)
ro o f
m a ss p a r tic ip a tio n fa c to r
ro o f le v e l a m p li tu d e
P F P a r tic ip a tio n F a c to r
Equivalent SDOF
Γ= mass participation factor m*= equivalent mass
𝑆𝑎 = 𝑉
𝑚∗Γ
𝑆𝑑 = Δ𝑢Γ
25
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
URM building in Yverdon-les-Bains
Static nonlinear analysis 26
Cas
e st
ud
y in
: Le
stu
zzi P
. an
d B
ado
ux
M. (
20
13
) Év
alu
atio
n p
aras
ism
iqu
e d
es c
on
stru
ctio
ns
exis
tan
tes
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Architectural plan
Static nonlinear analysis 27
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Simplification for the modeling
Static nonlinear analysis 28
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM)
Key-parameter: Frame effect (coupling of horizontal/vertical elements) – flexible slab (w/o frame effect)
– Infinitely stiff lintels
(total frame effect)
Static nonlinear analysis
0
2
3to ta l
h h
0
1
2s to r e y
h h
Zero
mo
me
nt
29
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Extreme case studies:
Static nonlinear analysis
0
26 .7
3to ta l
h h m
0
11 .2 5
2s to re y
h h m
30
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
Static nonlinear analysis
Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir Y]
Nxd [kN] 180 160 1200 360 1520 -
VRd [kN] 21.7 24.6 123.3 113.6 304.1 618
δu [%] 0.8 0.8 0.8 0.8 0.8 -
Δy [mm] 10.3 7.1 10 5.6 10.5 7.9
Δu [mm] 27.7 25.3 27.5 24.2 27.9 24.2
k [kN/mm] 2.1 3.5 12.4 20.2 29 77.8
31
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
𝑉𝑅𝑑, 𝑅 = 𝑙𝑤 ∙ 𝑁𝑥𝑑2 ∙ ℎ𝑤
1 − 1.15 ∙𝑁𝑥𝑑
𝑙𝑤 ∙ 𝑡𝑤 ∙ 𝑓𝑥𝑑
𝑉𝑅𝑑, 𝑆 = 1.5 ∙𝑓𝑣𝑑00.85 ∙ 𝑓𝑥𝑑
+ 0.4 ∙ 𝑁𝑥𝑑 ≈ 0.5 ∙ 𝑁𝑥𝑑
Static nonlinear analysis 32
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
∆𝑦 = 𝑉𝑅𝑑 ∙ 𝐻𝑡𝑜𝑡ℎ𝑝 ∙ 3 ∙ ℎ0 − ℎ𝑝6 ∙ 𝐸 ∙ 𝐼𝑒𝑓𝑓
+κ
𝐺 ∙ 𝐴𝑒𝑓𝑓
∆𝑦 = 𝛿𝑦 ∙ 𝐻𝑡𝑜𝑡
𝛿𝑦 =𝑑𝑦ℎ𝑝
∆𝑢 = ∆𝑦 + 𝑑𝑢 − 𝑑𝑦
𝛿𝑢 =𝑑𝑢ℎ𝑝 = 0.8%
Static nonlinear analysis 33
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
Static nonlinear analysis
0
100
200
300
400
500
600
700
0 5 10 15 20 25 30
Late
ral s
tren
gth
[kN
]
Top ultimate displacement dir Y [mm]
Lw = 2.0 m
Lw = 5.0 m
Lw = 6.0 m
Building (dir Y)
34
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
VRdy = (4*21.7+2*113.6+1*304) = 618 kN
kE = (4*2.1+2*20.2+1*29) = 77.8 kN / mm
Δy = 618 / 77.8 = 7.94 mm
Δu = 24.2 mm
Static nonlinear analysis 35
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
Static nonlinear analysis
Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir X]
Nxd [kN] 180 160 1200 360 1520 -
VRd [kN] 21.7 24.6 123.3 113.6 304.1 837.6
δu [%] 0.8 0.8 0.8 0.8 0.8 -
Δy [mm] 10.3 7.1 10 5.6 10.5 8.5
Δu [mm] 27.7 25.3 27.5 24.2 27.9 25.3
k [kN/mm] 2.1 3.5 12.4 20.2 29 98.6
36
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
Static nonlinear analysis
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25 30
Late
ral s
tren
gth
[kN
]
Top ultimate displacement [mm]
Lw = 2.4 m
Lw = 4.0 m
Building (dir X)
37
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
A) Frame effect not considered
VRdx = (4*123.3+14*24.6) = 837.6 kN
kE = (4*12.4+2*14*3.5) = 98.6 kN / mm
Δy = 837.6 / 98.6 = 8.50 mm
Δu = 25.3 mm
Static nonlinear analysis 38
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
Static nonlinear analysis
Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir Y]
Nxd [kN] 180 160 1200 360 1520 -
VRd [kN] 57.8 65.6 219.4 180.3 541.2 1133
δu [%] 0.8 0.8 0.8 0.4 0.8 -
Δy [mm] 10.2 7.6 12.0 6.5 14.5 9.8
Δu [mm] 27.6 25.7 29.0 14.9 30.9 14.9
k [kN/mm] 5.7 8.6 18.3 27.6 37.4 115.4
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25 30
Bas
e s
hea
r [k
N]
Top displacment [mm]
Lw = 2.4 m
Lw = 4.0 m
Building (dir X)
39
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
Static nonlinear analysis
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Late
ral s
tren
gth
[kN
]
Top ultimate displacement [mm]
Lw = 2.0 m
Lw = 5.0 m
Lw = 6.0 m
Building (dir Y)
40
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
VRdy = (4*57.8+2*180.3+1*541.2) = 1133 kN
kE = (4*5.7+2*27.6+1*37.4) = 115.4 kN / mm
Δy = 1133 / 115.4 = 9.81 mm
Δu = 14.9 mm
Static nonlinear analysis 41
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
Static nonlinear analysis
Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir X]
Nxd [kN] 180 160 1200 360 1520 -
VRd [kN] 57.8 65.6 219.4 180.3 541.2 1796
δu [%] 0.8 0.8 0.8 0.4 0.8 -
Δy [mm] 10.2 7.6 12.0 6.5 14.5 9.3
Δu [mm] 27.6 25.7 29.0 14.9 30.9 25.7
k [kN/mm] 5.7 8.6 18.3 27.6 37.4 193.6
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25 30
Bas
e s
hea
r [k
N]
Top displacment [mm]
Lw = 2.4 m
Lw = 4.0 m
Building (dir X)
42
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
Static nonlinear analysis
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 5 10 15 20 25 30 35
Late
ral S
tren
gth
[kN
Top ultimate displacement [mm]
Lw = 2.4 m
Lw = 4.0 m
Building (dir X)
43
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
B) Frame effect considered
VRdy = (4*219.4+14*65.6) = 1796 kN
kE = (4*18.3+14*8.6) = 193.6 kN / mm
Δy = 1796 / 193.6 = 9.28 mm
Δu = 25.7 mm
Static nonlinear analysis 44
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Static nonlinear analysis
Frame not considered Frame considered
45
Seismic safety factor obtained
Passing to ADSR format
𝑚∗ = 𝑚𝑖𝜱𝑖 (equivalent mass)
(1.0+0.75+0.50+0.25+0.10)*250 = 625 t = m*
Γ = 𝑚𝑖𝜱𝑖 𝑚𝑖𝜱𝑖2
=𝑚∗
𝑚𝑖𝜱𝑖2
(1.0+0.75+0.50+0.25+0.10)*(1.02+0.752+0.502+0.252+0.102)= 2.5/1.875 = 1.33 = Γ
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Static nonlinear analysis 46
Seismic safety factor obtained
Advanced Earthquake Engineering CIVIL-706
Unreinforced masonry (URM) Example
Static nonlinear analysis
dir VRd [KN] Say [m/s2]= VRd / m*· Γ
Δu/Γ [mm]
d* [mm]
α
Frame not considered
Y 618 0.74 18.2 14.4 1.26
X 838 1.01 19 12 1.58
Frame considered
Y 1133 1.36 11.2 9.9 1.13
X 1796 2.16 19.3 5.6 3.45
47
• DBV method = development of capacity curve through few geometrical and mechanical parameters. Method developed for masonry and reinforced concrete existing buildings.
MASONRY BUILDINGS
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 48
Pic
ture
s, f
orm
ula
s an
d g
rap
hs
in:
Luch
ini C
(2
01
6)
Ph
D T
he
sis.
D
eve
lop
me
nt
of
Dis
pla
cem
en
t-B
ase
d M
eth
od
s fo
r Se
ism
ic R
isk
Ass
ess
me
nt
of
the
Exis
tin
g B
uild
ing
Sto
ck
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
The equivalence between the multi degree of freedom (MDOF) and the single substitute one (SDOF) is established using the procedure proposed by Fajfar (2000) and assumed as reference also in Eurocode 8-Part 1 (2005), thus by introducing the Γ coefficient and the equivalent mass m*. The capacity curve assessment is associated with a certain analysis direction (dir = X,Y).
Static nonlinear analysis 49
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 𝑎𝑦,𝑑𝑖𝑟 =𝐹𝑑𝑖𝑟
𝑚∗Γ Yield acceleration
• 𝑇𝑦,𝑑𝑖𝑟 = 2𝜋𝑚∗
𝑘∗𝑑𝑖𝑟 Fundamental period
• 𝑑4 = 휀 ∙ 𝑑𝑢,𝑠.𝑠. 𝑑𝑖𝑟 + 1 − 휀 ∙ 𝑑𝑢,𝑢𝑛𝑖𝑓. 𝑑𝑖𝑟
Ultimate displacement capacity
Static nonlinear analysis 50
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 𝑎𝑦,𝑑𝑖𝑟 =𝐹𝑑𝑖𝑟
𝑚∗Γ Yield acceleration
𝐹𝑑𝑖𝑟 Total base shear capacity 𝑚∗ Equivalent mass of SDOF Γ Coefficient representing the modal participation factor and it requires of assuming a modal shape 𝜱
Γ = 𝑚𝑖𝜱𝑖 𝑚𝑖𝜱𝑖
2=𝑚∗
𝑚𝑖𝜱𝑖2
Static nonlinear analysis 51
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 𝑎𝑦,𝑑𝑖𝑟 =𝐹𝑑𝑖𝑟
𝑚∗Γ Yield acceleration
𝐹𝑑𝑖𝑟 Total base shear capacity Directly related to the shear strength offered by resistant walls area at the first floor
𝐹𝑑𝑖𝑟 = 𝐴1,𝑑𝑖𝑟 ∙ 𝜏𝑢,𝑑𝑖𝑟 ∙ 𝜉 ∙ 휁𝑟𝑒𝑠 𝜏𝑢,𝑑𝑖𝑟 Ultimate shear strength of masonry 𝜉 Coefficient aimed to penalize the strength as a function of the main prevailing failure mode expected at scale of masonry piers, it is assumed equal to 1 in the case of prevalence of shear failure mechanisms and 0.8 in the case of compression-bending failure mechanisms 휁𝑟𝑒𝑠 Corrective factor aimed to consider peculiarities of existing buildings
Static nonlinear analysis 52
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 𝑇𝑦,𝑑𝑖𝑟 = 2𝜋𝑚∗
𝑘∗𝑑𝑖𝑟
Fundamental period
𝑘∗𝑑𝑖𝑟 Stiffness of the SDOF system
𝑘∗𝑑𝑖𝑟 = 휁𝑟𝑖𝑔 ∙𝐺
𝐻2∙ 𝐴𝑖,𝑑𝑖𝑟 ∙ ℎ𝑖
𝑁
𝑖=1
휁𝑟𝑖𝑔 Corrective factor taking into account the coupling
effect due to spandrels and the flexural contribution
Static nonlinear analysis 53
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
54
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 휁𝑟𝑒𝑠 = 휁1휁2휁3
휁1 It takes into account the non homogeneous size of masonry piers
휁2 It takes into account the influence of geometric and shape irregularities in the plan configuration
휁3 It takes into account the influence of spandrels stiffness that directly affects the global collapse mechanism
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
55
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 휁𝑟𝑒𝑠 = 휁1휁2휁3
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
56
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 휁𝑟𝑖𝑔 = 휁4휁5
휁4 =1
1+1
1.2∙𝐺
𝐸∙ℎ𝑝
𝑏𝑝
2 coefficient aimed to take into account the influence of the
flexural component on the stiffness
ℎ𝑝 and 𝑏𝑝 height and width of masonry piers
휁5 coefficient related to the characteristics of spandrels
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
57
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
• 휁𝑟𝑒𝑠 = 휁1휁2휁3
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
58
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
• 𝑑4 = 휀 ∙ 𝑑𝑢,𝑠.𝑠. 𝑑𝑖𝑟 + 1 − 휀 ∙ 𝑑𝑢,𝑢𝑛𝑖𝑓. 𝑑𝑖𝑟
Ultimate displacement capacity
ultimate displacement of masonry piers according to a prevailing failure mode
δ𝑢,𝑑𝑖𝑟 ultimate drift of masonry piers according to a prevailing failure
mode in pier for the examined direction (if shear or flexural one)
𝑑𝑦,𝑑𝑖𝑟 yield displacement computed starting from 𝑎𝑦,𝑑𝑖𝑟 and 𝑇𝑦,𝑑𝑖𝑟
ε Fraction assigned to the soft story global failure mode
d1 = 0.7dy d2 = ρ2dy (2.15) where ρ2 is a coefficient which varies as a function of the prevailing global failure mode. It is proposed to assume a value equal to 1.5 in case of the soft story failure mode and 2 in case of the uniform one.
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Displacement based vulnerability method (DBV)
Static nonlinear analysis
where ζ1 takes into account the influence of the non homogeneous size of masonry piers; ζ2 the influence of geometric and shape irregularities in the plan configuration; ζ3 the influence of spandrels stiffness that directly affects the global collapse mechanism.
• 𝑑1 = 0.7𝑑𝑦
• 𝑑2 = ρ2𝑑𝑦
ρ2 coefficient varying as a function of the prevailing global failure mode. It is proposed to assume a value equal to 1.5 in case of soft story failure mode and 2 in case of the uniform one
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Displacement based vulnerability method (DBV)
Static nonlinear analysis
• The reliability of the new vulnerability models has been verified through a comparison between the capacity curves evaluated by using the mechanical models and the pushover obtained by non-linear static analyses.
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Displacement based vulnerability method (DBV)
Static nonlinear analysis 62
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Displacement based vulnerability method (DBV)
Static nonlinear analysis 63
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Displacement based vulnerability method (DBV)
Static nonlinear analysis 64
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 65
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 66
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 67
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 68
Advanced Earthquake Engineering CIVIL-706
Displacement based vulnerability method (DBV)
Static nonlinear analysis 69
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Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 70
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Seismic Damage
Static nonlinear analysis 71
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Seismic Damage
• Damage-based design/assessment:
– Related to social and economical costs of damage/mitigating measures
Static nonlinear analysis 72
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
EMS98 damage grades
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Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 74
Advanced Earthquake Engineering CIVIL-706
Response Spectrum
Static nonlinear analysis
Sa
Sd
ADSR spectrum
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Response Spectrum
Static nonlinear analysis 76
Advanced Earthquake Engineering CIVIL-706
Response Spectrum
Static nonlinear analysis
𝑆𝑑 ≈ 𝑆𝑎𝜔𝑛2
𝜔𝑛 ≈ 𝑘
𝑚
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Acceleration-Displacement Response Spectrum (ADRS)
Static nonlinear analysis 78
Advanced Earthquake Engineering CIVIL-706
Acceleration-Displacement Response Spectrum (ADRS)
Static nonlinear analysis 79
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SIA norm ADSR
Static nonlinear analysis
Sa Sd
TB < T < TC constant acceleration = agd * 3 factor (2.5; S; η)
T < TB increasing acceleration (tending for T=0 to agd * S)
T < TC decreasing acceleration
T < TD constant displacment
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SIA norm ADSR
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Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis
𝑆𝑎 = 𝑎𝑔𝑑 ∙ 𝑆 ∙ 1 +2.5휂 − 1 ∙ 𝑇
𝑇𝐵 0 ≤ 𝑇 ≤ 𝑇𝐵
𝑆𝑎 = 2.5 ∙ 𝑎𝑔𝑑 ∙ 𝑆 ∙ η 𝑇𝐵 ≤ 𝑇 ≤ 𝑇𝐵
𝑆𝑎 = 2.5 ∙ 𝑎𝑔𝑑 ∙ 𝑆 ∙ η ∙𝑇𝐶𝑇 𝑇𝐶 ≤ 𝑇 ≤ 𝑇𝐷
𝑆𝑎 = 2.5 ∙ 𝑎𝑔𝑑 ∙ 𝑆 ∙ η ∙𝑇𝐶 ∙ 𝑇𝐷𝑇2 𝑇𝐷 ≤ 𝑇
SIA norm ADSR
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Content
• Pushover technique – capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 83
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Determination of the performance point
Two of the main static non linear methods to determine the performance point (examples covered here)
• Equivalent linearization (FEMA 440) • Improved Spectrum Method of ATC40
• N2 method (equal displacement rule, EC8 approach)
Static nonlinear analysis 84
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Equivalent Linearization Method
A little bit of background…
• First introduced in 1970 in a pilot project as a
rapid evaluation tool (Freeman et al. 1975)
• Basis of the simplified analysis methodology in ATC-40 (1996)
• Improved later in FEMA 440 document (2005)
Static nonlinear analysis 85
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Equivalent Linearization Method
Based on equivalent linearization.
The displacement demand of a non-linear SDOF system is estimated from the displacement demand of a linear-elastic SDOF system. The elastic SDOF system, referred to as an equivalent system, has a period and a damping ratio larger than those of the initial non-linear system (ATC, 2005).
Static nonlinear analysis
A version of the Capacity-Spectrum Method (CSM) is proposed by ATC (Applied Technology Council, US). This version is based on equivalent linearization. Therefore, the displacement demand of a non-linear SDOF system is estimated from the displacement demand of a linear-elastic SDOF system. The elastic SDOF system, referred to as an equivalent system, has a period and a damping ratio larger than those of the initial non-linear system (ATC, 2005).
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Equivalent Linearization
Static nonlinear analysis
Basic equations…
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Equivalent Linearization- Performance Point
Static nonlinear analysis
Sou
rce:
FEM
A 4
40
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 휂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 휂
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Equivalent Linearization- Performance Point
Static nonlinear analysis 89
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 휂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 휂
𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 Spectral displacement of the equivalent system
𝑇𝑒𝑞 Equivalent period of vibration
ζ𝑒𝑞 Equivalent viscous damping ratio
𝑆𝑑 𝑇𝑒𝑞; ζ=5% Displacement demand of the linear system with 5%-damping elastic
ratio 휂 Reduction factor depending from the damping modification factor
휂 =1
0.5+10ζ𝑒𝑞
S_d (T_eq;ξ_eq )
휂 =1
0.5 + 10 𝜉𝑒𝑞
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Equivalent Linearization- Performance Point
Static nonlinear analysis 90
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 휂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 휂
The equivalent period and the equivalent damping ratio are functions of the strength reduction factor of the non-linear SDOF system and, respectively, of the initial period of vibration and of the damping ratio. The various equivalent linear methods differ from each other mainly for functions used to compute 𝑇𝑒𝑞 and ζ𝑒𝑞.
In their work (2008), Lin & Miranda give the equivalent period and the equivalent damping ratio as follows:
𝑇𝑒𝑞 = 1 +𝑚1𝑇2∙ 𝑅𝜇
1.8 − 1 ∙ 𝑇
ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2∙ 𝑅𝜇 − 1
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Equivalent Linearization- Performance Point
Static nonlinear analysis 91
𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 휂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2
4𝜋2∙ 휂
𝑇𝑒𝑞 = 1 +𝑚1𝑇2∙ 𝑅𝜇
1.8 − 1 ∙ 𝑇
ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2∙ 𝑅𝜇 − 1
Advanced Earthquake Engineering CIVIL-706
Equivalent Linearization
• Advantages:
– Linear computation
– Use of pushover analysis
• Drawbacks:
– Value of damping
– Not always conservative
Static nonlinear analysis 92
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N2 Method
A little bit of background…
• Started in the mid 1980’s (Fajfar and Fischinger 1987, 1989)
• A variant of the Capacity Spectrum Method (ATC-40)
• Based on inelastic spectra rather than elastic spectra
Static nonlinear analysis 93
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N2 Method Procedure
Advanced Earthquake Engineering CIVIL-706
N2 Method Procedure
Static nonlinear analysis
Inelastic response spectrum (ADRS)
,a e
a d d e
SS S S
R R
( 1) 1C
C
C
TR T T
T
R T T
Reduction factor
“Vidic et al. 1992”
95
Ductility factor
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 96
Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 97
N2 Method Procedure
Advanced Earthquake Engineering CIVIL-706
N2 Method Procedure (3)-Performance Point
Static nonlinear analysis
Basic equations…
:C
if T T
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N2 Method Procedure (3)-Performance Point
Static nonlinear analysis
*1 ( 1)
d e C
d
S TS R
R T
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N2 Method Procedure (3)-Performance Point
Static nonlinear analysis
Basic equations…
:C
if T T
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N2 Method Procedure (3)-Performance Point
Static nonlinear analysis
d deS S
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N2 Method Procedure
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Content
• Capacity curve
• Seismic damage
• Acceleration-Displacement Response Spectrum (ADRS)
• Performance Point
• Large-scale vulnerability assessment
Static nonlinear analysis 103