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

CE-409: Introduction to Structural Dynamics and Earthquake Engineering

MODULE 7

Prof. Dr. Akhtar Naeem Khan & Prof. Dr. Mohammad Javed [email protected] [email protected]

1

University of Engineering & Technology, Peshawar, Pakistan

RESPONSE OF INELASTIC SDOF SYSTEMS TO EARTHQUAKE LOADING

CE-409: MODULE 7 (Fall-2013) 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).



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


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 reduce by a factor to obtain design forces.



Earthquake-Resistant and NOT Earthquake-Proof: Damage is expected during an earthquake in normal constructions (a) undamaged building, and (b) damaged building.

Structures must have sufficient initial stiffness to ensure the non- occurrence 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.


Two aspects are worth consideration: 1.Force carrying ability under seismic demand2. Ability to absorb energy under seismic demandNote: Compromise can be made on force carrying ability


CE-409: MODULE 7 (Fall-2013)


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.



8

Ductility: Buildings are designed and detailed to develop favorable failure mechanisms that possess: 1.specified lateral strength, 2.reasonable stiffness and, above all,3. good post-yield deformability.

Sharp reduction in strength w/o significant displacements after peak strength

CE-409: MODULE 7 (Fall-2013) 9999

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) 10

Construction of Design Spectrum (firm soil)Acceleration sensitive region

Velocity sensitive region Displacement sensitive region

CE-409: MODULE 7 (Fall-2013) 1111111111

Figure: Pseudo- acceleration design spectrum (84.1 th percentile) drawn on linear scale for ground motions with ;

ζ = 1,2,5,10 and 20 %.

in. 36u and in/sec, 48u,1u gogogo === g

Design Spectrum for various values of ζ

2.71

CE-409: MODULE 7 (Fall-2013) 12

Comparison of base shear coefficients from elastic design spectrum and International Building Code 2000.

R=1.5

R=8

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

CE-409: MODULE 7 (Fall-2013) 13131313

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

Response of Elastoplastic SDOF system to Earthquake loading

Lateral force, fS

Lateral displacement, u

fs

u

CE-409: MODULE 7 (Fall-2013) 141414

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

Elastoplastic idealization of a non-linear system

Force-deformation curve : actual and elastoplastic idealization based on equal energy principle

k

CE-409: MODULE 7 (Fall-2013) 15

Elastic system corresponding to a given elastoplastic system

um = peak deformations in elastoplastic system; and, uo = peak deformation

in the corresponding linear elastic system when both are subjected to same ground motion.

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

ufs

CE-409: MODULE 7 (Fall-2013) 16

Normalized yield strength of an elastoplastic system,

The normalized yield strength of an elastoplastic system is defined as:

yf

o

y

y f

ff =

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

yf

CE-409: MODULE 7 (Fall-2013) 17

Normalized yield strength of an elastoplastic system,

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, of a system is less than 1.0, the system will deform beyond its linearly elastic limit. e.g., = 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.

yf

oyy /fff =

yf

75.0== oyy /fff

CE-409: MODULE 7 (Fall-2013) 18

o

y

o

y

o

yy u

u

ku

ku

f

ff ===yf can also be expressed as:

Normalized yield strength of an elastoplastic system, yf

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) 19

Yield strength reduction factor, Ry

fy can also be related to fo through a yield strength reduction factor,

Ry as:

In other words

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 strength required for the system to remain elastic divided by 2.

i.e .,

y

o

y

oy u

u

f

fR ==

y

o

yy

o

y ff

ff

fR

11 ===

2o

y

ff =

CE-409: MODULE 7 (Fall-2013) 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.

y

m

u

u=µ

fs

u

fy

uyum

CE-409: MODULE 7 (Fall-2013) 21

Relation b/w μ and yf

y

y

o

y ff

uf .

u

u

u

u

u

1

u

u

f

f

y

m

o

m

yo

y

o

y =⇒=⇒==

yy

o

m

Rf

u

u µµ ==⇒ .

This relationship couples the peak displacements of elastoplastic (um) and corresponding elastic (uo) system

CE-409: MODULE 7 (Fall-2013) 22

Elastoplastic system under cyclic loading

Elastoplastic force-deformation relation

+fs-fs

a-b-c =+ve loading, c-d = unloading, d-e-f = -ve loading , f-g= unloading, g-h= +ve loading

a

b c

d

ef

g

h

CE-409: MODULE 7 (Fall-2013) 23

a

b c

d

ef

g

h

Loading Unloading

ReloadingUnloading Reloading

dc

f g

Elastoplastic system under cyclic loadinga b c+fs

g h+fsf

de - fs

a-b-c c-d

d-e-ff-g g-h

CE-409: MODULE 7 (Fall-2013) 24

Equation of motion for elastoplastic system

(t)umfucum gs −=++

The EOM for an elastic SDOF system subjected to ground motion is:

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:

)u(u,ff ss =

(t)um)u(u,fucum gs −=++⇒

CE-409: MODULE 7 (Fall-2013) 25

Equation of motion for elastoplastic system

(t)u)u(u,fm

um

cu gs −=++⇒ 1

Substituting )u(u,ff

)u(u,fmc s

y

sn

~ and 2 == ωζ

(t)u)fu(u,fm

um

mu gys

n −=++ ~12 ωζ

yny umukfs )(. ince 2y ω==

(t)u)u(u,fuuu gsynn −=++⇒ ~2 2ωζω

CE-409: MODULE 7 (Fall-2013) 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

deformation response history of the system subjected to El Centro ground motion is shown in the below given figure.

Tn=0.5 sec, ζ=0

u

g,ug

0.319ggou =

CE-409: MODULE 7 (Fall-2013) 27

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 system, /gu)m/w/(Aw/Amku/w/wf t

s−====

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

Minimum strength required for a system to remain linear elastic

CE-409: MODULE 7 (Fall-2013) 28

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

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

0.125/ffor 0.125foyy

==0.171w1.37w*0.125f 0.125f

oy===⇒

Effect of yielding on deformation response history of elastoplastic system

CE-409: MODULE 7 (Fall-2013) 29

Response of elastoplastic system with Tn = 0.5 sec, ζ = 0, and to El-Centro ground motion: (a) deformation; (b) resisting force and acceleration; (c) time intervals of yielding; (d) force–deformation relation.

Effect of yielding on deformation response history of elastoplastic system

125.0f y =

10 sec

CE-409: MODULE 7 (Fall-2013) 30

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

To keep the discussion simple at this stage, it is assumed that the elastoplastic systems considered in discussion can indefinitely yield in plastic range.

implies a linearly elastic system

0.125 and 0.25 0.5,

0.1y =f

Effect of on deformation response history of elastoplastic system

yf

,0.1y =f

oyy /fff =

CE-409: MODULE 7 (Fall-2013) 31


yfu

, in

0.1=y

f

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

a

b c

d

ef

g

h

a

b c

d

ef

g

h

oy ff =

CE-409: MODULE 7 (Fall-2013) 32

u, i

nEffect of on deformation response history of

elastoplastic systemyf

5.0=yf

a

b c

d

ef

g

h

of

of5.0=yf

Typical +ve loading-unloading and -ve loading-unloading for a single cycle

CE-409: MODULE 7 (Fall-2013) 33


yf

a

b c

d

ef

g

h

of

of25.0=yf

Typical +ve loading-unloading and -ve loading-unloading for a single cycle

25.0=yf

CE-409: MODULE 7 (Fall-2013) 34


yf

125.0=yf

a

b c

d

ef

g

h

of

of125.0=yf Typical +ve loading-unloading and -ve loading-unloading for a single cycle

CE-409: MODULE 7 (Fall-2013) 35

up-in

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

Tn=0.5 sec, ζ=5% and peak value of disp. in elastic SDOF system=uo=2.25"

yo

m

fu

u 1.=µ.inum −yf up

up is the residual displacement in the elastoplastic system at the end of ground motion. up=0 in case of elastic system

Effect of on ductility demand, μ, and residual deformation, up , of elastoplastic system

yf

El-centro 1940

CE-409: MODULE 7 (Fall-2013) 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 = 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.

yf

CE-409: MODULE 7 (Fall-2013) 37

Ductility demand, μD

The ductility demand for the system with , Tn=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) 38

Effect of Tn on ductility demand, μD

yy Rfμ === 1/8.0

yy Rfμ === 1/4.0

yy Rfμ === 1/2.0

yy Rfμ === 1/1.0

CE-409: MODULE 7 (Fall-2013) 39

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 Tn and

approximately equal to Ry (i.e. )

For systems with Tn in velocity sensitive region (intermediate to

long structures) the ductility demand may be larger or smaller than 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

Effect of Tn on ductility demand, μ

y/1 f

yf

CE-409: MODULE 7 (Fall-2013) 40

Construction of constant ductility response spectrum

uy corresponding to various values of are determined by

the equation:

Constant ductility response spectrum for Dy is drawn using Dy

= uy.

Design spectrums for Vy (Pseudo-velocity response spectrum)

and Ay (Pseudo-acceleration response spectrum) can be

constructed using the relations:

y

2

n

n2

yyy

n

nyyD

T

2π.ωDA & D

T

2π.ωDV

==

==

yf

yoy

o

y

yf.u uor

u

uf ==

CE-409: MODULE 7 (Fall-2013) 41

Relations b/w and yield strength ,fy , and base shear coefficient for elasto plastic system, Ay/g

( ) ( )wf

f

y

yy

.g

AA.

g

w

mAuωmumωk.u

y

y

yn

2

yn

2

y

==⇒

====

g

A

w

f yy =⇒

CE-409: MODULE 7 (Fall-2013) 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) 43

Combined Inelastic Dy-Vy-Ay response spectrum for constant ductility factors

CE-409: MODULE 7 (Fall-2013) 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:

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

Tf=Tf ′.

><<−

<

= TT μ

TTT 12μ

TT 1

R

cn

c'nb

an

y

CE-409: MODULE 7 (Fall-2013) 45

Inelastic Pseudo-velocity design spectrum (New-mark Hall)Td=Td ′ as V and D are divided by same value of Ry i.e. μ

Tc≠Tc ′ as V and A are divided by different values value of Ry

CE-409: MODULE 7 (Fall-2013) 46

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

CE-409: MODULE 7 (Fall-2013) 47

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

yn

yy VT

2πVA

== nω

Once Vy is calculated by Vy=V/μ, then Ay can be be easily calculated by:

CE-409: MODULE 7 (Fall-2013) 48

Inelastic Pseudo-acceleration design spectrum (Newmark-Hall)-normal scale

CE-409: MODULE 7 (Fall-2013) 49

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

ymμuu =

Where

It is already known

( )( ) n

2

y

n2

yy

y ω

A

mω

mA

k

fu ===

2

n

yy 2π

TAuor

=

y

2

n

ymA

2π

Tμμuu

==⇒

CE-409: MODULE 7 (Fall-2013) 50

Inelastic peak deformation design spectrum (Newmark-Hall)-log scale

yn

nyy VT

2πVA

== ω

In order to draw inelastic deformation design spectrum, inelastic peak deformations, um, is calculated using following relation (slide 49)

y

2n

m A2

Tu

=

πµ

where Ay is calculated by equation mentioned on slide 47 i.e

CE-409: MODULE 7 (Fall-2013) 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.

.wg

Af y

y=

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:

For above determined value of Ay, the corresponding value of fy is 0.49gw/g = 0.49w

CE-409: MODULE 7 (Fall-2013) 52

0.49g

Application of the Inelastic design spectrum: 1. Structural design for allowable ductility

CE-409: MODULE 7 (Fall-2013) 53

The peak deformation, um, can be related to Ay as follows

( )( )

yyy

o

y A

A

/g.wA

A/g.w

f

fR ===

A2π

Tμu

y

2

n

m

=We have already derived the relation

yR

AA y =⇒

A2π

T

R

μuor

2

n

y

m

=

Please recall that A is the elastic pseudo- acceleration

Application of the Inelastic design spectrum: 1. Structural design for allowable ductility

CE-409: MODULE 7 (Fall-2013) 54

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

Application of the Inelastic design spectrum: 2. Evaluation of existing structures

CE-409: MODULE 7 (Fall-2013) 55

Force-displacement curve of a building using Pushover analysis?

Draw a sketch of frame with plastic hinges


Collapse

First plastic hinge

fs

u u

fy

fs

umuy

CE-409: MODULE 7 (Fall-2013) 56

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:

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

peak deformation um, can be determined

by using eqn. derived on slide 52

gw

fAor .w

g

Af y

y

y

y

==

yyA/AR and =

A2π

T

R

μu

2

n

y

m

=


CE-409: MODULE 7 (Fall-2013) 57

Problem M7.1Consider 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) 58

Problem M7.1 (contd…..)

Slide 54

Slide 56 A2π

T

R

μu

2

n

y

m

=

CE-409: MODULE 7 (Fall-2013) 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) 60


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) 61


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) 62

It is a common practice in earthquake engineering to indicate structural damages in terms of performance levels.

Performance levels as per FEMA 356?

Operational

(O)

Very light damages(Building can be

occupied. No repair work required)

Immediate occupancy

(IO)

Light damages(Building can be

occupied but will need repair work)

Life safety

(LS)

Moderate damages(Building can be

occupied after subsequent repair)

Collapse prevention

(CP)

Severe damages(Building is far beyond

the economically feasible repair)


CE-409: MODULE 7 (Fall-2013) 63IO performance levelO performance level

CP performance level

LS performance level

O

IO

LS

CP


CE-409: MODULE 7 (Fall-2013) 64


The inelastic design spectrum is also useful for direct Displacement-based 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) 65


CE-409: MODULE 7 (Fall-2013) 66

Problem M 7.2: Consider a long reinforced-concrete 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) 67

Problem M 7.2 contd….

CE-409: MODULE 7 (Fall-2013) 68


CE-409: MODULE 7 (Fall-2013) 69


CE-409: MODULE 7 (Fall-2013) 70


CE-409: MODULE 7 (Fall-2013) 71

Home Assignment No. 6

Solve problems 7.7 and 7.8 from chopra’s book