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University of Engineering & Technology,
Peshawar, Pakistan

CE-409: Introduction to Structural Dynamics and
Earthquake Eng...
BASIC ASPECTS OF SEISMIC DESIGN
Designing buildings to behave elastically during earthquakes without
damage may render the...
BASIC ASPECTS OF SEISMIC DESIGN

Earthquake-Resistant Design Philosophy for buildings:
(a)Minor (Frequent) Shaking – No/Ha...
BASIC ASPECTS OF SEISMIC DESIGN
Buildings are designed only for a fraction of the force that they
would experience, if the...
BASIC ASPECTS OF SEISMIC DESIGN
Structures must have sufficient initial stiffness to ensure the nonoccurrence of structura...
BASIC ASPECTS OF SEISMIC DESIGN
Two aspects are worth consideration:
1.Force carrying ability under seismic demand
2. Abil...
BASIC ASPECTS OF SEISMIC DESIGN
The design for only a fraction of the elastic level of seismic
forces is possible, only if...
BASIC ASPECTS OF SEISMIC DESIGN
Sharp reduction in strength w/o
significant displacements after
peak strength

Ductility: ...
BASIC ASPECTS OF SEISMIC DESIGN
Peak base shear induced in a linearly elastic system by ground
motion is Vb = (A/g)w. wher...
Construction of Design Spectrum (firm soil)
Acceleration sensitive region
Velocity sensitive region

CE-409: MODULE 7 (Fal...
Design Spectrum for various values of ζ

2.71

Figure: Pseudo- acceleration design spectrum (84.1 th percentile) drawn on ...
BASIC ASPECTS OF SEISMIC DESIGN
A/g= 0.4* (2.71g) =1.09

R, is detailed in the slides to
follow. This
is a factor
which pr...
Response of Elastoplastic SDOF system to
Earthquake loading
In this lecture, we will study the earthquake response of
elas...
Elastoplastic idealization of a non-linear system

fy=
Lateral force at which
yielding
start in idealized
elastoplastic
sy...
Elastic system corresponding to a given elastoplastic
system
u
fs

Elastoplastic system
and
corresponding elastic system h...
Normalized yield strength of an elastoplastic
system, f y
The normalized yield strength f y of
an elastoplastic system is ...
Normalized yield strength of an elastoplastic
system, f y
fo can be interpreted as the strength required for the structure...
Normalized yield strength of an elastoplastic
system, f y
f y ku y u y
f y can also be expressed as: f y =
=
=
f o kuo uo
...
Yield strength reduction factor, Ry
fy can also be related to fo through a yield strength reduction factor,
Ry as:

f o uo...
Displacement ductility factor
Ductility factor,μ, of an elastoplastic system is defined as the ratio of
peak (or absolute ...
Relation b/w μ and f y

fy

uy

fy
1
um um
fy = =
⇒
=
⇒
=
. fy
fo u o
uo u y
uo uy
um
µ
⇒
= µ. f y =
uo
Ry
This relationsh...
Elastoplastic system under cyclic loading

-fs

+fs

b

c
h

a

g

f

d

e

Elastoplastic force-deformation relation
a-b-c...
Elastoplastic system under cyclic loading
a

+fs

c

b

b

d

c

c

h
a

g

f

d

Loading

f

d-e-f

c-d

a-b-c

e

e

d

...
Equation of motion for elastoplastic system
The EOM for an elastic SDOF system subjected to ground
motion is:

 

mu ...
Equation of motion for elastoplastic system

c
1




⇒u + u + f s (u,u) = −u g (t)
m
m
Substituting


f s (u,u) ~
...
Minimum strength required for a
system to remain linear elastic
Consider an elastic SDOF system with
weight w, Tn=0.5 sec,...
Minimum strength required for a system to remain
linear elastic
Time variation of fs/w (i.e ratio of elastic resisting for...
Effect of yielding on deformation response history of
elastoplastic system
Now consider an elastoplastic SDOF system with ...
Effect of yielding on deformation response history of
elastoplastic system
10 sec

Response of elastoplastic system with T...
Effect of f y on deformation response history of
elastoplastic system
Now we examine how the response of elastoplastic sys...
Effect of f y on deformation response history of
elastoplastic system

f = 1.0
u, in

y

f y = fo
h

h
a

a

g
f

c

b

c
...
Effect of f y on deformation response history of
elastoplastic system
u, in

f y = 0.5

fo
f y = 0 .5 f o

c

b
h
a

g
f

...
Effect of f y on deformation response history of
elastoplastic system

f y = 0.25

fo

f y = 0.25 f o

c

b
h

Typical +ve...
Effect of f y on deformation response history of
elastoplastic system

f y = 0.125

fo

Typical +ve loading-unloading and
...
Effect of f y on ductility demand, μ, and residual
deformation, up , of elastoplastic system

um − in.

um 1
µ=
.
uo f y

...
Ductility demand, μD
The values of μ as calculated on previous slide are known as
ductility demand.
Thus the ductility dem...
Ductility demand, μD
The ductility demand for the system with

, T n=0.5 sec

and ζ =5% was found to be 3.11 when subjecte...
Effect of Tn on ductility demand, μD

μ = 8.0 = 1/f y = R y
μ = 4.0 = 1/f y = R y
μ = 2.0 = 1/f y = R y
μ = 1.0 = 1/f y = ...
Effect of Tn on ductility demand, μ
Following observations can be made from the figure given on
previous slide.
For system...
Construction of constant ductility response spectrum
uy corresponding to various values of f are determined by
y

u
f =
or...
Relations b/w and yield strength ,fy , and base
shear coefficient for elasto plastic system, Ay/g

f = k.u = ( mω ) u = m(...
Inelastic pseudo-acceleration response spectrum for
constant ductility factors

Effect of Tn on fy/w (i.e. yielding
base s...
Combined Inelastic Dy-Vy-Ay response spectrum
for constant ductility factors

CE-409: MODULE 7 (Fall-2013)

43
Inelastic Pseudo-velocity design spectrum
The very first step in the construction of inelastic design spectrum for
constan...
Inelastic Pseudo-velocity design spectrum (New-mark Hall)
Tc≠Tc ′ as V and A are divided
by different values value of Ry

...
Inelastic design spectra (Newmark-Hall) for firm
soil with PGA=1g

CE-409: MODULE 7 (Fall-2013)

46
Inelastic Pseudo-acceleration design spectrum
(Newmark-Hall)-log scale

Once Vy is calculated by Vy=V/μ, then
Ay can be be...
Inelastic Pseudo-acceleration design spectrum
(Newmark-Hall)-normal scale

CE-409: MODULE 7 (Fall-2013)

48
Relation between um and Ay determined from inelastic
pseudo- acceleration design spectrum

u = μu

It is already known

m
...
Inelastic peak deformation design spectrum
Hall)-log scale

(Newmark-

In order to draw inelastic deformation design spect...
Application of the Inelastic design spectrum:
1. Structural design for allowable ductility
Consider a SDOF system having a...
Application of the Inelastic design spectrum:
1. Structural design for allowable ductility

0.49g

CE-409: MODULE 7 (Fall-...
Application of the Inelastic design spectrum:
1. Structural design for allowable ductility
The peak deformation, um, can b...
Application of the Inelastic design spectrum:
2. Evaluation of existing structures
Consider the simplest possible structur...
Application of the Inelastic design spectrum:
2. Evaluation of existing structures Draw a sketch of
frame with plastic
hin...
Application of the Inelastic design spectrum:
2. Evaluation of existing structures
For a system with known Tn and ζ, A is ...
Problem M7.1
Consider a one-story frame with lumped weight w, Tn = 0.25 sec, and fy
= 0.512w. Assume that ζ = 5% and elast...
Problem M7.1 (contd…..)

Slide 54

2

Slide 56

μ T 
u =
 A
R  2π 
n

m

y

CE-409: MODULE 7 (Fall-2013)

58
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
Displacement Based...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
A simple example o...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
After earthquakes ...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
It is a common pra...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
LS
IO

CP

O

CP p...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
The inelastic desi...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures

CE-409: MODULE 7 ...
Application of the Inelastic design spectrum:
3. Direct displacement based seismic design of structures
Problem M 7.2: Con...
Problem M 7.2 contd….

CE-409: MODULE 7 (Fall-2013)

67
Problem M 7.2 contd….

CE-409: MODULE 7 (Fall-2013)

68
Problem M 7.2 contd….

CE-409: MODULE 7 (Fall-2013)

69
Problem M 7.2 contd….

CE-409: MODULE 7 (Fall-2013)

70
Home Assignment No. 6

Solve problems 7.7 and 7.8 from chopra’s book

CE-409: MODULE 7 (Fall-2013)

71
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Module 7 (RESPONSE OF INELASTIC S.D.O.F SYSTEMS TO EARTHQUAKE LOADING)

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Transcript of "Module 7 (RESPONSE OF INELASTIC S.D.O.F SYSTEMS TO EARTHQUAKE LOADING)"

  1. 1. University of Engineering & Technology, Peshawar, Pakistan CE-409: Introduction to Structural Dynamics and Earthquake Engineering MODULE 7 RESPONSE OF INELASTIC SDOF SYSTEMS TO EARTHQUAKE LOADING Prof. Dr. Akhtar Naeem Khan & drakhtarnaeem@nwfpuet.edu.pk Prof. Dr. Mohammad Javed mjaved@nwfpuet.edu.pk 1
  2. 2. BASIC ASPECTS OF SEISMIC DESIGN Designing buildings to behave elastically during earthquakes without damage may render the project economically unviable. As a consequence, The design philosophy for earthquake resistant design of structure is to allow damage and thereby dissipate the energy input to it during the earthquake. Therefore, the traditional earthquake-resistant design philosophy requires that normal buildings should be able to resist: (a)Minor and frequent shaking with no/un-notable damage to structural and non-structural elements; (b) Moderate shaking with minor to moderate damage (repairable) to structural and non-structural elements; and (c) Severe and infrequent shaking with damage to structural elements, but with NO collapse (to save life and property inside/adjoining the building). CE-409: MODULE 7 (Fall-2013) 2
  3. 3. BASIC ASPECTS OF SEISMIC DESIGN Earthquake-Resistant Design Philosophy for buildings: (a)Minor (Frequent) Shaking – No/Hardly any damage, (b) Moderate Shaking – Minor to moderate structural damage, and (c) Severe (Infrequent) Shaking – Structural damage, but NO collapse CE-409: MODULE 7 (Fall-2013) 3
  4. 4. BASIC ASPECTS OF SEISMIC DESIGN Buildings are designed only for a fraction of the force that they would experience, if they were designed to remain elastic during the expected strong ground shaking (see given below figure) , and thereby permitting damage (inelastic range) see figure on next slide. Basic strategy of earthquake design: Calculate maximum elastic forces and CE-409: MODULE 7 (Fall-2013) reduce by a factor to obtain design forces. 4
  5. 5. BASIC ASPECTS OF SEISMIC DESIGN Structures must have sufficient initial stiffness to ensure the nonoccurrence of structural damage under minor shaking. Thus, seismic design balances reduced cost and acceptable damage, to make the project viable. For this reason, design against earthquake effects is called as earthquake-resistant design and not earthquake-proof design. Earthquake-Resistant and NOT Earthquake-Proof: Damage is expected during an earthquake in normal constructions (a) undamaged building, and (b) damagedCE-409: MODULE 7 (Fall-2013) building. 5
  6. 6. BASIC ASPECTS OF SEISMIC DESIGN Two aspects are worth consideration: 1.Force carrying ability under seismic demand 2. Ability to absorb energy under seismic demand Note: Compromise can be made on force carrying ability CE-409: MODULE 7 (Fall-2013) 6
  7. 7. BASIC ASPECTS OF SEISMIC DESIGN The design for only a fraction of the elastic level of seismic forces is possible, only if the building can stably withstand large displacement demand through structural damage without collapse and undue loss of strength. This property is called ductility (see Figure on next slide). It is relatively simple to design structures to possess certain lateral strength and initial stiffness by appropriately proportioning the size and material of the members. But, achieving sufficient ductility is more involved and requires extensive laboratory tests on full-scale specimen to identify preferable methods of detailing. CE-409: MODULE 7 (Fall-2013)
  8. 8. BASIC ASPECTS OF SEISMIC DESIGN Sharp reduction in strength w/o significant displacements after peak strength Ductility: Buildings are designed and detailed to develop favorable failure mechanisms that possess: 1.specified lateral strength, 2.reasonable stiffness and, above all, CE-409: deformability. 3. good post-yield MODULE 7 (Fall-2013) 8
  9. 9. BASIC ASPECTS OF SEISMIC DESIGN Peak base shear induced in a linearly elastic system by ground motion is Vb = (A/g)w. where w is the weight of the system and A is the pseudo acceleration corresponding to the natural vibration period and damping of the system. Most buildings (as already discussed) are designed, however, for base shear smaller than the elastic base shear ,Vb = (A/g)w This becomes clear from figure on next slide, where the base shear coefficient A/g from the design spectrum of Fig. 6.9.5(chopra’s book), scaled by 0.4 to correspond to peak ground acceleration of 0.4g, is compared with the base shear coefficient specified in the 2000 International Building Code CE-409: MODULE 7 (Fall-2013) 9
  10. 10. Construction of Design Spectrum (firm soil) Acceleration sensitive region Velocity sensitive region CE-409: MODULE 7 (Fall-2013) Displacement sensitive region 10
  11. 11. Design Spectrum for various values of ζ 2.71 Figure: Pseudo- acceleration design spectrum (84.1 th percentile) drawn on linear scale for ground motions with  go = 1g , u go = 48 in/sec, and u go = 36 in. ;  u ζ = 1,2,5,10 and CE-409: MODULE 7 (Fall-2013) 20 %. 11 11
  12. 12. BASIC ASPECTS OF SEISMIC DESIGN A/g= 0.4* (2.71g) =1.09 R, is detailed in the slides to follow. This is a factor which primarily depend upon material , structural system and detailing and is used to work out the design base shear. In the literature, it is generally referred to as force reduction factor for a structural system R=1.5 R=8 Comparison of base shear coefficients from elastic design spectrum and International Building Code 2000. CE-409: MODULE 7 (Fall-2013) 12
  13. 13. Response of Elastoplastic SDOF system to Earthquake loading In this lecture, we will study the earthquake response of elastic-perfectly plastic ( referred as elastoplastic systems) SDOF systems to earthquake motions. Note that elastoplastic system is an idealized response of a non-linear system u fs Lateral force, fS Lateral displacement, u CE-409: MODULE 7 (Fall-2013) 13
  14. 14. Elastoplastic idealization of a non-linear system fy= Lateral force at which yielding start in idealized elastoplastic system. Also known as yield strength uy= Yield displacement in elastoplastic system. It is also called yield deformation um = Maximum displacement in idealized elastoplastic system Note That initial stiffness, k, of both the systems must be same k Force-deformation curve : actual and elastoplastic idealization based on equal energy principle CE-409: MODULE 7 (Fall-2013) 14 14
  15. 15. Elastic system corresponding to a given elastoplastic system u fs Elastoplastic system and corresponding elastic system has the same stiffness. Similarly both systems have same mass and damping. Consequently, natural vibration period, Tn, of elastoplastic system and corresponding elastic system is the same as long as u ≤ uy. k, m and ζ are same for the two systems um = peak deformations in elastoplastic system; and, uo = peak deformation in the corresponding linear elastic system when both are subjected to same ground motion. CE-409: MODULE 7 (Fall-2013) 15
  16. 16. Normalized yield strength of an elastoplastic system, f y The normalized yield strength f y of an elastoplastic system is defined as: f f = f y y o Where fo and uo are the peak values of force and deformation, respectively, in the linear elastic system corresponding to elastoplastic system under the same ground motion. For brevity the notation fo is used instead of fso (elastic resisting force) as followed in previous lectures CE-409: MODULE 7 (Fall-2013) 16
  17. 17. Normalized yield strength of an elastoplastic system, f y fo can be interpreted as the strength required for the structure to remain within its linear elastic limit during the ground motion. If the normalized yield strength, f y = f y /f o of a system is less than 1.0, the system will deform beyond its linearly elastic limit. e.g., f y = 0.75 implies that the yield strength of the elastoplastic system is 0.75 times the strength required for the system to remain elastic during the ground motion. f y = f y /f o = 0.75 CE-409: MODULE 7 (Fall-2013) 17
  18. 18. Normalized yield strength of an elastoplastic system, f y f y ku y u y f y can also be expressed as: f y = = = f o kuo uo Please note again that uy is the displacement at which yielding start in the elastoplastic system . Whereas, uo is the peak displacement in the corresponding elastic system . This uo must not be confused with the maximum displacement in the elastoplastic system, um CE-409: MODULE 7 (Fall-2013) 18
  19. 19. Yield strength reduction factor, Ry fy can also be related to fo through a yield strength reduction factor, Ry as: f o uo Ry = = f y uy In other words f 1 1 R = = = f f f f o y y y y o Ry is greater than 1 for a system that deforms into inelastic range. Ry=2 implies that the yield strength of the elastoplastic system is the strengthfrequired for the system to remain elastic divided by 2. i.e ., fy = o 2 CE-409: MODULE 7 (Fall-2013) 19
  20. 20. Displacement ductility factor Ductility factor,μ, of an elastoplastic system is defined as the ratio of peak (or absolute maximum) deformation to the yield deformation. um µ= uy fs fy uy u CE-409: MODULE 7 (Fall-2013) um 20
  21. 21. Relation b/w μ and f y fy uy fy 1 um um fy = = ⇒ = ⇒ = . fy fo u o uo u y uo uy um µ ⇒ = µ. f y = uo Ry This relationship couples the peak displacements of elastoplastic (um) and corresponding elastic (uo) system CE-409: MODULE 7 (Fall-2013) 21
  22. 22. Elastoplastic system under cyclic loading -fs +fs b c h a g f d e Elastoplastic force-deformation relation a-b-c =+ve loading, c-d = unloading, d-e-f = -ve loading , f-g= unloading, g-h= +ve loading CE-409: MODULE 7 (Fall-2013) 22
  23. 23. Elastoplastic system under cyclic loading a +fs c b b d c c h a g f d Loading f d-e-f c-d a-b-c e e d f g - fs Unloading +fs Unloading Reloading f-g CE-409: MODULE 7 (Fall-2013) g h Reloading g-h 23
  24. 24. Equation of motion for elastoplastic system The EOM for an elastic SDOF system subjected to ground motion is:    mu + cu + f s = − mu g (t) Force fs corresponding to deformation u, in case of inelastic system, is not single valued and depends upon the history of deformations and on whether the deformation is increasing (positive velocity) or decreasing (negative velocity) see Figure on slide 22. Thus the resisting force fs in case of inelastic system can be expressed as:  f s = f s (u,u)     ⇒ mu + cu + f s (u,u) = −mu g (t) CE-409: MODULE 7 (Fall-2013) 24
  25. 25. Equation of motion for elastoplastic system c 1     ⇒u + u + f s (u,u) = −u g (t) m m Substituting  f s (u,u) ~  c = 2ζmωn and = f s (u,u) fy 2ζmωn 1 ~     u+ u + f s (u,u)f y = −u g (t) m m since f y = k .u y = (ωn 2 m)u y ~     ⇒ u + 2ζωn u + ωn u y f s (u,u) = −u g (t) 2 CE-409: MODULE 7 (Fall-2013) 25
  26. 26. Minimum strength required for a system to remain linear elastic Consider an elastic SDOF system with weight w, Tn=0.5 sec, and ζ=0. The u Tn=0.5 sec, ζ=0 deformation response history of the system subjected to El Centro ground motion is  ug , g shown in the below given figure.  u go = 0.319g CE-409: MODULE 7 (Fall-2013) 26
  27. 27. Minimum strength required for a system to remain linear elastic Time variation of fs/w (i.e ratio of elastic resisting force to the weight of system) for the system on previous slide is shown. For an undamped f s /w = ku/w = mA / w = A /( w / m) = − t /g u system, It can be seen that fo/w=1.37 or fo=1.37w . i.e., The minimum strength required for the structure (Tn=5% and ζ=0) to remain elastic (when subjected to 1940 El-centro earthquake) is 1.37w CE-409: MODULE 7 (Fall-2013) 27
  28. 28. Effect of yielding on deformation response history of elastoplastic system Now consider an elastoplastic SDOF system with same properties (as given on previous slide i.e.,w, Tn=0.5 sec, and ζ=0) and with a normalized yield strength of f = 0.125 or f /f = 0.125 y y o ⇒ f = 0.125 f = 0.125 *1.37w = 0.171w y o The deformation response history of the system (developed using EOM for elastoplastic systems given at the end of slide 25) subjected to El- Centro ground motion is shown on the next slide for first 10 sec. The peak displacement also occur in first 10 seconds CE-409: MODULE 7 (Fall-2013) 28
  29. 29. Effect of yielding on deformation response history of elastoplastic system 10 sec Response of elastoplastic system with Tn = 0.5 sec, ζ = 0, and f y = 0.125 to ElCE-409: MODULE 7 (Fall-2013) Centro ground motion: (a) deformation; (b) resisting force and acceleration; (c) time 29
  30. 30. Effect of f y on deformation response history of elastoplastic system Now we examine how the response of elastoplastic system is affected by its yield strength. Consider four SDOF systems all with identical properties in their linear elastic range (i.e Tn=0.5 sec and ζ=5%) but with different normalized yield strengths of f y = 1.0, 0.5, 0.25 and 0.125 To keep the discussion simple at this stage, it is assumed that the elastoplastic systems considered in discussion can indefinitely yield in f y range. plastic = 1.0 implies a linearly elastic system f y = f y /f o CE-409: MODULE 7 (Fall-2013) 30
  31. 31. Effect of f y on deformation response history of elastoplastic system f = 1.0 u, in y f y = fo h h a a g f c b c b g d e f d e um=maximum displacement in elastoplastic system subjected to El-centro 1940 ground motion, up = permanent/residual displacement in the elastoplastic system (at the end of El-centro 1940 ground motion). up=0 in case of elastic system CE-409: MODULE 7 (Fall-2013) 31
  32. 32. Effect of f y on deformation response history of elastoplastic system u, in f y = 0.5 fo f y = 0 .5 f o c b h a g f d e Typical +ve loading-unloading and -ve loading-unloading for a single cycle CE-409: MODULE 7 (Fall-2013) 32
  33. 33. Effect of f y on deformation response history of elastoplastic system f y = 0.25 fo f y = 0.25 f o c b h Typical +ve loading-unloading and -ve loading-unloading for a single cycle a g f CE-409: MODULE 7 (Fall-2013) d e 33
  34. 34. Effect of f y on deformation response history of elastoplastic system f y = 0.125 fo Typical +ve loading-unloading and -ve loading-unloading for a single cycle f y = 0.125 f o c b h a g d CE-409: MODULE 7 (Fall-2013) f e 34
  35. 35. Effect of f y on ductility demand, μ, and residual deformation, up , of elastoplastic system um − in. um 1 µ= . uo f y 1.00 2.25 1.00 0 0.50 1.62 1.44 0.17 0.25 1.75 3.11 1.1 0.125 2.07 7.36 1.13 fy up-in Tn=0.5 sec, ζ=5% and peak value of disp. in elastic SDOF system=uo=2.25" up El-centro 1940 up is the residual displacement in the elastoplastic system at the end of ground motion. up=0 in case of elastic system CE-409: MODULE 7 (Fall-2013) 35
  36. 36. Ductility demand, μD The values of μ as calculated on previous slide are known as ductility demand. Thus the ductility demand imposed by El Centro ground motion on inelastic systems having f y = 0.5,0.25,0.125 are 1.44, 3.11 and 7.36 respectively. Ductility demand represents a requirement on the design in the sense that the ductility capacity, previously defined as displacement ductility factor (i.e., the ability to deform beyond the elastic limit) should exceed the ductility demand. CE-409: MODULE 7 (Fall-2013) 36
  37. 37. Ductility demand, μD The ductility demand for the system with , T n=0.5 sec and ζ =5% was found to be 3.11 when subjected to El-Centro ground motion. A system with above mentioned properties and having ductility capacity greater than 3.11 will survive collapse when subjected to El Centro 1940 ground motion. However, another system with same properties but having a ductility capacity of 3 will collapse when subjected to El-Centro 1940ground motion It may be noted that μ is used for displacement ductility factor (i.e ductility capacity) as well as ductility demand in the text book being followed. However, we will follow μD for ductility demand and μC for ductility capacity. Note, here μ will be taken as ductility factor CE-409: MODULE 7 (Fall-2013) 37
  38. 38. Effect of Tn on ductility demand, μD μ = 8.0 = 1/f y = R y μ = 4.0 = 1/f y = R y μ = 2.0 = 1/f y = R y μ = 1.0 = 1/f y = R y CE-409: MODULE 7 (Fall-2013) 38
  39. 39. Effect of Tn on ductility demand, μ Following observations can be made from the figure given on previous slide. For systems with Tn in displacement sensitive region (long structures) the ductility demand is independent of T n and approximately equal to R (i.e. 1 / f y ) y For systems with Tn in velocity sensitive region (intermediate to long structures) the ductility demand may be larger or smaller than fy Ry; and the influence of , although small, is not negligible. For systems with Tn in acceleration sensitive region (short structures) the ductility demand much be much larger than Ry, specially in case of very short structures CE-409: MODULE 7 (Fall-2013) 39
  40. 40. Construction of constant ductility response spectrum uy corresponding to various values of f are determined by y u f = or u = u .f u the equation: y y y o y o Constant ductility response spectrum for Dy is drawn using = uy. Dy Design spectrums for Vy (Pseudo-velocity response spectrum) and Ay (Pseudo-acceleration response spectrum) can be constructed using the relations: 2  2π   2π  V = D .ω =   D & A = D .ω =   D T  T      2 y y n y y n CE-409: MODULE 7 (Fall-2013) y n y n 40
  41. 41. Relations b/w and yield strength ,fy , and base shear coefficient for elasto plastic system, Ay/g f = k.u = ( mω ) u = m( ω u ) = mA 2 y ⇒ 2 n y n y y y A w f = .A = .w g g y y y f A ⇒ = w g y y CE-409: MODULE 7 (Fall-2013) 41
  42. 42. Inelastic pseudo-acceleration response spectrum for constant ductility factors Effect of Tn on fy/w (i.e. yielding base shear coefficient is insignificant when Tn≥ 1.5 sec CE-409: MODULE 7 (Fall-2013) 42
  43. 43. Combined Inelastic Dy-Vy-Ay response spectrum for constant ductility factors CE-409: MODULE 7 (Fall-2013) 43
  44. 44. Inelastic Pseudo-velocity design spectrum The very first step in the construction of inelastic design spectrum for constant ductility is to develop the elastic design spectrum using procedure explained in previous lecture Once the elastic design spectrum is developed, the inelastic design spectrum for constant ductility is obtained by dividing its various branches by Ry (details given on next slide). One of the proposal suggested by Newmark and Hall (Figure 7.11.3) for correlating Ry with Tn is: 1 Tn < Ta   R y =  2μ −1 Tb < Tn < Tc' μ Tn > Tc  Where Ta,Tb,……… are mentioned on the inelastic design spectra given on next slide. It must be noted that Ta=Ta′ ,Tb=Tb ′, Td=Td ′, Te=Te ′ and CE-409: MODULE 7 (Fall-2013) 44 T =T ′.
  45. 45. Inelastic Pseudo-velocity design spectrum (New-mark Hall) Tc≠Tc ′ as V and A are divided by different values value of Ry CE-409: MODULE 7 (Fall-2013) Td=Td ′ as V and D are divided by same value of Ry i.e. μ 45
  46. 46. Inelastic design spectra (Newmark-Hall) for firm soil with PGA=1g CE-409: MODULE 7 (Fall-2013) 46
  47. 47. Inelastic Pseudo-acceleration design spectrum (Newmark-Hall)-log scale Once Vy is calculated by Vy=V/μ, then Ay can be be easily calculated by:  2π  A y = Vyωn =   T Vy   n CE-409: MODULE 7 (Fall-2013) 47
  48. 48. Inelastic Pseudo-acceleration design spectrum (Newmark-Hall)-normal scale CE-409: MODULE 7 (Fall-2013) 48
  49. 49. Relation between um and Ay determined from inelastic pseudo- acceleration design spectrum u = μu It is already known m ( mA ) = A f u = = k ( ω m) ω y Where y y y 2 y 2 n T  or u = A    2π  n 2 n y y 2 T  ⇒ u = μu = μ   A  2π  n m y CE-409: MODULE 7 (Fall-2013) y 49
  50. 50. Inelastic peak deformation design spectrum Hall)-log scale (Newmark- In order to draw inelastic deformation design spectrum, inelastic peak deformations, um, is calculated using following relation (slide 49) 2 um  Tn  = µ  Ay  2π  where Ay is calculated by equation mentioned on slide 47 i.e  2π  A y = Vyωn =  Vy T   n CE-409: MODULE 7 (Fall-2013) 50
  51. 51. Application of the Inelastic design spectrum: 1. Structural design for allowable ductility Consider a SDOF system having allowable ductility,μ, which is decided on the ductility capacity of the material and design details selected It is desired to determine the design yield strength, fy, and the design deformation, um, for the system. For the known values of Tn, ζ , μ the value of Ay/g is determined of from Figure 7.11.5 or 7.11.6 (Chopra’s book) given on slides 46 and 47. e.g., Ay=0.49g for Tn=1 sec, ζ=5% and μ=4 as shown on next slide. The required yield strength is determined from relation: A f = .w g y y For above determined value of Ay, the corresponding value of fy is 0.49gw/g = 0.49w CE-409: MODULE 7 (Fall-2013) 51
  52. 52. Application of the Inelastic design spectrum: 1. Structural design for allowable ductility 0.49g CE-409: MODULE 7 (Fall-2013) 52
  53. 53. Application of the Inelastic design spectrum: 1. Structural design for allowable ductility The peak deformation, um, can be related to Ay as follows ( A/g.w ) = A f R = = ( A /g.w ) A f o y A ⇒ Ay = Ry y y y Please recall that A is the elastic pseudoacceleration 2 T  We have already derived the relation u = μ  A  2π  n m y 2 μ T  or u =  A R  2π  n m y CE-409: MODULE 7 (Fall-2013) 53
  54. 54. Application of the Inelastic design spectrum: 2. Evaluation of existing structures Consider the simplest possible structures, SDF system, having mass m, initial stiffness k at small displacement. The yield strength fy of the structure are determined from its properties: dimensions, member sizes, and design details (reinforcement in R.C. structures, connections in steel structures). fy can be determined from any suitable method from existing analytical methods based on extensive laboratory works. Result of a pushover analysis? for determining fy is shown on next slide. Tn for small oscillation is computed from k and m, and the damping ratio ζ from field tests CE-409: MODULE 7 (Fall-2013) 54
  55. 55. Application of the Inelastic design spectrum: 2. Evaluation of existing structures Draw a sketch of frame with plastic hinges First plastic hinge fs Collapse fs fy u u uy um Force-displacement curve of a building using Pushover analysis? CE-409: MODULE 7 (Fall-2013) 55
  56. 56. Application of the Inelastic design spectrum: 2. Evaluation of existing structures For a system with known Tn and ζ, A is read from elastic design spectrum Ay for known value of fy and Ry can be determined using: A f f = .w or A = g w   g  and R = A/A  y y y y y y With Tn already known, μ for calculated value of Ry can be determined by using the applicable equation determined from three 3 equations given on slide 44 μ  Tn peak deformation um, can be determined by using eqn. derived on slide 52 CE-409: MODULE 7 (Fall-2013) 2  u =  A R  2π  m y 56
  57. 57. Problem M7.1 Consider a one-story frame with lumped weight w, Tn = 0.25 sec, and fy = 0.512w. Assume that ζ = 5% and elastoplastic force–deformation behavior. Determine the lateral deformation for the design earthquake has a peak acceleration of 0.5g and its elastic design spectrum is given by Fig. 6.9.5 multiplied by 0.5 Solution: For a system with Tn = 0.25 sec, A = (2.71g)0.5 = 1.355g from Fig. 6.9.5 (slide 10) Slide 41 Slide 53 CE-409: MODULE 7 (Fall-2013) 57
  58. 58. Problem M7.1 (contd…..) Slide 54 2 Slide 56 μ T  u =  A R  2π  n m y CE-409: MODULE 7 (Fall-2013) 58
  59. 59. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures Displacement Based Seismic Design (DBSD) is defined broadly as any seismic design in which displacement related quantities are used directly to judge performance acceptability. This performance acceptability for various limit states/performance levels in general is referred to as Performance Based Seismic Design (PBD) among earthquake engineering community A simple DBSD approach could be to specify a drift limit corresponding to a defined damage level, and then require that the drift under the specified seismic loading does not exceed the specified drift. This procedure is in contrast with Force Based Seismic Design (FBSD) procedure in which the acceptability of structural performance is judged on the basis of force –based quantities. CE-409: MODULE 7 (Fall-2013) 59
  60. 60. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures A simple example of a force-based procedure is the familiar requirement that the design base shear strength under seismic loading shall not be less than some fraction of base shear calculated assuming linear elastic structural response. We followed FBSD process in previous slides of this module due to the reason that currently seismic codes are based on FBSD procedure because of their familiarity for design against other loading such as gravity and wind. CE-409: MODULE 7 (Fall-2013) 60
  61. 61. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures After earthquakes of 1994 Northridge, USA and 1995 Kobe, Japan, earthquake engineering community is seriously making effort to PBD (see figure on next slide) which is essentially based on DBSD procedure. It is worth mentioning that neither of the two procedures (i.e., FBSD and DBSD) can be totally taken independent of the decision making parameters involved in the two procedures. Inherently, both involves relevant parameters related to forces and displacements, however, decisions are based on Forces in FBSD and Displacements/Drifts in DBSD. CE-409: MODULE 7 (Fall-2013) 61
  62. 62. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures It is a common practice in earthquake engineering to indicate structural damages in terms of performance levels. Performance levels as per FEMA 356? Immediate occupancy (O) Life safety Collapse prevention (LS) Operational (CP) (IO) Moderate damages Very light damages (Building can be occupied. No repair work required) Light damages (Building can be occupied but will need repair work) Severe damages (Building can be occupied after subsequent repair) (Building is far beyond the economically feasible repair) CE-409: MODULE 7 (Fall-2013) 62
  63. 63. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures LS IO CP O CP performance level CE-409: MODULE 7 (Fall-2013) O performance level IO performance level LS performance level 63
  64. 64. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures The inelastic design spectrum is also useful for direct Displacementbased design of structures. The goal is to determine the initial stiffness and yield strength of the structure necessary to limit the deformation to some acceptable value. Applied to an elastoplastic SDF system (Fig. 7.12.1), such a design procedure may be implemented as a sequence of the following steps: 1. Estimate the yield deformation uy for the system. 2. Determine acceptable plastic rotation θp of the hinge at the base. 3. Determine the design displacement um from um = uy + hθp and design ductility factor from CE-409: MODULE 7 (Fall-2013) 64
  65. 65. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures CE-409: MODULE 7 (Fall-2013) 65
  66. 66. Application of the Inelastic design spectrum: 3. Direct displacement based seismic design of structures Problem M 7.2: Consider a long reinforcedconcrete viaduct that is part of a freeway. The total weight of the superstructure, 13 kips/ft, is supported on identical bents 30 ft high, uniformly spaced at 130 ft. Each bent consists of a single circular column 60 in. in diameter (Fig. E7.3a). Using the displacement-based design procedure, design the longitudinal reinforcement of the column for the design earthquake has a peak acceleration of 0.5g and its elastic design spectrum is given by Fig. 6.9.5 multiplied by 0.5 CE-409: MODULE 7 (Fall-2013) 66
  67. 67. Problem M 7.2 contd…. CE-409: MODULE 7 (Fall-2013) 67
  68. 68. Problem M 7.2 contd…. CE-409: MODULE 7 (Fall-2013) 68
  69. 69. Problem M 7.2 contd…. CE-409: MODULE 7 (Fall-2013) 69
  70. 70. Problem M 7.2 contd…. CE-409: MODULE 7 (Fall-2013) 70
  71. 71. Home Assignment No. 6 Solve problems 7.7 and 7.8 from chopra’s book CE-409: MODULE 7 (Fall-2013) 71
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