STATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering ... · PDF fileSTATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering ... analysis Non linear ... Force method Response

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


Earthquake Engineering Assessment

Action Structure

Static Dynamic

Linear Linear static

analysis Linear dynamic

analysis

Non-linear Non linear static

analysis Non linear

dynamic analysis


Action Structure

Static Dynamic

Linear Equivalent Force

method Response

Spectrum method

Non-linear Pushover Non-linear Dynamic

4


Linear static analysis

Action Structure

Static Dynamic


method Response

Spectrum method



• 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


Equivalent Force Method

Action Structure

Static Dynamic


method Response

Spectrum method



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


Equivalent Force Method

Action Structure

Static Dynamic


method Response

Spectrum method


Static nonlinear analysis 7


Linear dynamic analysis

Action Structure

Static Dynamic


method Response

Spectrum method



• 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



Action Structure

Static Dynamic


method Response

Spectrum method



• Response spectrum method

• Overlapping of effects (SRSS: square root of the sum of the square)

9



Action Structure

Static Dynamic


method Response

Spectrum method





Action Structure

Static Dynamic


method Response

Spectrum method





Action Structure

Static Dynamic


method Response

Spectrum method




Non linear time history analysis

• Advantage: the complexity of the dynamic load is considered + the dynamic behavior of structure

• Disadvantage: time consuming



Non linear static procedure

• How does it work? Capacity vs. Demand


Sa

Sd

14

• Necessary elements?

– Acceleration-displacement response spectrum (ADRS)

– Capacity Curve


Nonlinear static procedure


V (force)

Δ (disp.)

Sa

Sd

15



• Developing capacity curves:

– Displacement-based methods




• 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



Content

• Capacity curve

• Seismic damage

• Acceleration-Displacement Response Spectrum (ADRS)

• Performance Point

• Large-scale vulnerability assessment



Content

• Capacity curve

• Seismic damage






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


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


Capacity Curve

• Pushover

• Simplified analytical method (EC8)

• DBV method


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


Capacity Curve


• Pushover

Load pattern: importance definition

– Effect of lateral load can change the capacity curve

Base Shear


Capacity Curve

• Pushover


– Effect of lateral load pattern


• Pushover

Different model type


Capacity Curve

• Pushover


– Effect of lateral load pattern


Structural model (FEM, AEM, etc.)

24


Capacity Curve

• Force/Displacement to spectral coordinates


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


Unreinforced masonry (URM) Example

URM building in Yverdon-les-Bains


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



Architectural plan




Simplification for the modeling



Unreinforced masonry (URM)

Key-parameter: Frame effect (coupling of horizontal/vertical elements) – flexible slab (w/o frame effect)

– Infinitely stiff lintels

(total frame effect)


0

2

3to ta l

h h

0

1

2s to r e y

h h

Zero

mo

me

nt

29



Extreme case studies:


0

26 .7

3to ta l

h h m

0

11 .2 5

2s to re y

h h m

30



A) Frame effect not considered


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




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





∆𝑦 = 𝑉𝑅𝑑 ∙ 𝐻𝑡𝑜𝑡ℎ𝑝 ∙ 3 ∙ ℎ0 − ℎ𝑝6 ∙ 𝐸 ∙ 𝐼𝑒𝑓𝑓

+κ

𝐺 ∙ 𝐴𝑒𝑓𝑓

∆𝑦 = 𝛿𝑦 ∙ 𝐻𝑡𝑜𝑡

𝛿𝑦 =𝑑𝑦ℎ𝑝

∆𝑢 = ∆𝑦 + 𝑑𝑢 − 𝑑𝑦

𝛿𝑢 =𝑑𝑢ℎ𝑝 = 0.8%






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




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






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





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




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




B) Frame effect considered


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





0

200

400

600

800

1000

1200

0 5 10 15 20 25 30 35

Late

ral s

tren

gth

[kN

]


Lw = 2.0 m

Lw = 5.0 m

Lw = 6.0 m

Building (dir Y)

40




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






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





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


Lw = 2.4 m

Lw = 4.0 m

Building (dir X)

43




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





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




Seismic safety factor obtained




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


Displacement based vulnerability method (DBV)


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



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).




• 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






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






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




• 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





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



• 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



55






56



• 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



57






58





• 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.

59





• 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

60




• 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.

61


























Content

• Capacity curve

• Seismic damage






Seismic Damage



Seismic Damage

• Damage-based design/assessment:

– Related to social and economical costs of damage/mitigating measures


Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis

EMS98 damage grades

73


Content

• Capacity curve

• Seismic damage






Response Spectrum


Sa

Sd

ADSR spectrum

75


Response Spectrum



Response Spectrum


𝑆𝑑 ≈ 𝑆𝑎𝜔𝑛2

𝜔𝑛 ≈ 𝑘

𝑚

77


Acceleration-Displacement Response Spectrum (ADRS)



Acceleration-Displacement Response Spectrum (ADRS)



SIA norm ADSR


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

80


SIA norm ADSR

81


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

82


Content

• Pushover technique – capacity curve

• Seismic damage






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)



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)



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).


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).

86


Equivalent Linearization


Basic equations…

87


Equivalent Linearization- Performance Point


Sou

rce:

FEM

A 4

40

𝑆𝑑 = 𝑆𝑑 𝑇𝑒𝑞; ζ𝑒𝑞 = 𝑆𝑑 𝑇𝑒𝑞; ζ=5% ∙ 휂 = 𝑆𝑎 𝑇𝑒𝑞; ζ=5% ∙𝑇𝑒𝑞2

4𝜋2∙ 휂

88





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





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





4𝜋2∙ 휂

𝑇𝑒𝑞 = 1 +𝑚1𝑇2∙ 𝑅𝜇

1.8 − 1 ∙ 𝑇

ζ𝑒𝑞 = ζ=5% +𝑛1𝑇𝑛2∙ 𝑅𝜇 − 1


Equivalent Linearization

• Advantages:

– Linear computation

– Use of pushover analysis

• Drawbacks:

– Value of damping

– Not always conservative



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


Advanced Earthquake Engineering CIVIL-706 Static nonlinear analysis 94

N2 Method Procedure


N2 Method Procedure


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



N2 Method Procedure


N2 Method Procedure (3)-Performance Point


Basic equations…

:C

if T T

98




*1 ( 1)

d e C

d

S TS R

R T

99




Basic equations…

:C

if T T

100




d deS S

101


N2 Method Procedure


Content

• Capacity curve

• Seismic damage





Documents

STATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering ... · PDF fileSTATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering ... analysis Non linear ... Force method Response