Professional Publications, Inc. • Belmont, California

Seismic Design ofBuilding StructuresA Professional’s Introduction toEarthquake Forces and Design DetailsEleventh Edition

Michael R. Lindeburg, PEwith Kurt M. McMullin, PhD, PE

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5 Response of Structures

1. Elastic Response Spectra . . . . . . . . . . . . . . . . . 5-12. Idealized Response Spectra . . . . . . . . . . . . . . . 5-13. Response Spectra for Other Earthquakes . . 5-34. Log Tripartite Graph . . . . . . . . . . . . . . . . . . . . 5-35. Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36. Strain Energy and Ductility Factor . . . . . . . 5-47. Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-58. Large Ductility Swings . . . . . . . . . . . . . . . . . . . 5-59. Inelastic Response Spectra . . . . . . . . . . . . . . . 5-5

10. Normalized Design Response Spectra . . . . . . 5-611. Response Spectrum for

IBC-Defined Soil Profiles . . . . . . . . . . . . . . . 5-612. Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-613. P-� Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-714. Torsional Shear Stress . . . . . . . . . . . . . . . . . . . 5-815. Negative Torsional Shear . . . . . . . . . . . . . . . . 5-1016. Overturning Moment . . . . . . . . . . . . . . . . . . . . 5-1117. Rigid Frame Buildings . . . . . . . . . . . . . . . . . . . 5-1118. High-Rise Buildings . . . . . . . . . . . . . . . . . . . . . 5-11

1. ELASTIC RESPONSE SPECTRA

The response of a building to earthquake ground motiondepends on the dynamic characteristics of the building.Specifically, the natural period (see Sec. 3.8) and thedamping ratio (see Sec. 4.8) affect building responsemore than do other factors. For a given damping ratio,�, and for a given ground motion, a curve known as aresponse spectrum of spectral acceleration, Sa, can bedrawn that plots the maximum acceleration response ofan elastic single-degree-of-freedom system against thenatural period of the system. The response spectrumfor a particular earthquake can be used to determinethe theoretical maximum acceleration response of thebuilding.

A family of curves (i.e., response spectra) for an actualearthquake for various damping ratios is illustrated inFig. 5.1. Similar response spectra can be developed forspectral velocity and spectral displacement.

The spectra shown in Fig. 5.1 are for elastic response toan earthquake. That is, the structures used to developthe curves moved and swayed during the earthquake,but there was no yielding. For that reason, the curvesare known as elastic response spectra.

There will always be a region on the response spectrumwhere the acceleration is highest. This occurs where thenatural building period coincides with the predominantearthquake period—when the building is in resonance

with the earthquake. For California earthquakes, thepeak usually occurs in the 0.2 sec to 0.5 sec periodrange.1 Theoretically, infinite resonant response (i.e.,an infinite magnification factor) is possible, though itis highly unlikely since all real structures are damped.2

It seems intuitively logical that a building with largeamounts of internal damping will resist acceleration(i.e., motion) to a greater extent than will a similarbuilding with no damping. Such behavior is actuallyobserved as spectral acceleration decreases becausedamping increases, although the effect of damping atlower periods is slight (since the natural periods ofundamped and lightly damped structures are essentiallythe same).

2. IDEALIZED RESPONSE SPECTRA

The response spectra derived from the behavior of oneSDOF system in one particular earthquake are usuallyquite jagged, as shown in Fig. 5.1. It is not possible touse such a historical record for design, since it is unlikelythat an earthquake matching the original earthquake induration, magnitude, or time history will occur. Also,even if the design earthquake was completely specified,the significant variation in spectral values over smallperiod ranges would require an unreasonable accuracyin the determination of the building period. To getaround these problems, a smoothed average design

1This is not always the case, as shown by the Loma Prieta earthquake.2A properly designed and constructed building seldom experiences trueresonance. Planned or unplanned yielding occurs before true resonantresponse is achieved, and this yielding damps out the resonance.

Figure 5.1 Typical Elastic Response Spectra (1940 El Centroearthquake in N-S direction)

1 2 3 4 5

1000

2000

Sa(cm/s2)

natural period (sec)

ζ = 0%

5%

10%

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response spectrum based on the envelopes of perfor-mance of several earthquakes is developed, as illustratedin Fig. 5.2.

Example 5.1

The primary support for an industrial drill press with amass of 100,000 lbm (45 000 kg) is the structural steelbent shown. The beam-column and base connections arerigid. The horizontal beam has a mass of 119 lbm/ft(160 kg/m), neglecting the weight of the vertical sup-ports. The system has 5% damping. Determine the elas-tic response (i.e., base shear) for a 1940 El Centroearthquake in the north-south direction. Use averagedesign spectra.

W14 × 82 each column

W18 × 119

E = 2.9 × 107 psi (2 × 105 MPa)

I = 882 in4 each (3.67 × 10–4 m4 each)

10 ft (3 m) 10 ft (3 m)

N

22 ft(6.6 m)

100,000 lbm (45 000 kg)

Customary U.S. Solution

The total mass, m, of the moving system is

m¼ 100;000 lbmþ ð20 ftÞ 119lbmft

� �¼ 102;380 lbm

From Table 4.1, the combined stiffness, kt, of the twovertical supports is

kt ¼ ð2Þ 12EI

h3

� �

¼ð2Þð12Þ 2:9� 107

lbfin2

� �ð882 in4Þ

ð22 ftÞ3 12inft

� �2

¼ 4� 105 lbf=ft

From Eq. 4.14 and Eq. 4.15, the natural period of vibra-tion, T, is

T ¼ 2pffiffiffiffiffiffiffimgck

r

¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi102;380 lbm

32:2ft-lbmlbf-sec2

� �4� 105

lbfft

� �vuut

¼ 0:56 sec

From Fig. 5.2, the spectral acceleration for this periodand 5% damping is Sa=5.5 ft/sec2. From Eq. 3.1, thebase shear is

V ¼ mSa

gc¼

ð102;380 lbmÞ 5:5ftsec2

� �32:2

ft-lbmlbf-sec2

¼ 1:75� 104 lbf

SI Solution

The total mass, m, of the moving system is

m ¼ 45 000 kgþ ð6 mÞ 160kg

m

� �¼ 45 960 kg

From Table 4.1, the combined stiffness, kt, of the twovertical supports is

kt ¼ ð2Þ 12EI

h3

� �

¼ð2Þð12Þð2� 105 MPaÞ 106

PaMPa

� �ð3:67� 10�4 m4Þ

ð6:6 mÞ3

¼ 6:13� 106 N=m

Figure 5.2 Average Elastic Design Response Spectra (based onthe 1940 El Centro earthquake) [multiplier =1]

ζ = 0%

2%

5%10%

20%

40%

0.4 0.8 1.2 1.6 2.0 2.4 2.8

2

6

8

10

12

14

16

18

20

22

4

natural period (sec)

max

imu

m a

ccel

erat

ion

(ft

/sec

2 )

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5-2 S E I S M I C D E S I G N O F B U I L D I N G S T R U C T U R E S

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From Eq. 4.14 and Eq. 4.15, the natural period of vibra-tion, T, is

T ¼ 2pffiffiffiffiffimk

q¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi45 960 kg

6:13� 106Nm

vuut¼ 0:54 s

From Fig. 5.2, the spectral acceleration for this periodand 5% damping is approximately Sa=1.68 m/s2. FromEq. 3.1, the base shear is

V ¼ mSa ¼ ð45 960 kgÞ 1:68ms2

� �¼ 7:72� 104 N

3. RESPONSE SPECTRA FOR OTHEREARTHQUAKES

The design response spectra in Fig. 5.2, although normal-ized and averaged over several earthquakes, are adjustedfor an earthquake of a specific magnitude and peakground acceleration. Based on historical data and prob-ability studies, the recurrence interval for an earthquakeof that magnitude can be determined. For example, anearthquake of the 1940 El Centro magnitude is expectedat that site, on the average, every 70 years. However,smaller earthquakes will be experienced more frequentlythan every 70 years, and larger earthquakes will beexperienced less frequently than every 70 years.

In order to apply the average design response spectra toother earthquakes, they are simply scaled upward ordownward for larger and smaller earthquakes, respec-tively. For example, Table 5.1 gives the scale factor(to be used to scale Fig. 5.2 downward) for other recur-rence intervals at the El Centro site.

4. LOG TRIPARTITE GRAPH

Since spectral acceleration, velocity, and displacementfor linear elastic response are all related (see Eq. 3.2), allthree spectral quantities can be shown by a single curveon a graph with three different scales. Such a graph isdone on a logarithmic scale and is known as a logtripartite plot. Both elastic and inelastic (see Sec. 5.9)tripartite plots are prepared. However, for inelasticresponse, the spectral acceleration, velocity, and dis-placement cannot be represented by a single curve onthe tripartite plot.

Tripartite plots, both elastic and inelastic, can differ inhow the axes are arranged. Figure 5.3 illustrates twocommon arrangements for presenting the information,while Fig. 5.4 gives an elastic log tripartite plot for the1940 El Centro earthquake.

5. DUCTILITY

The expected magnitude of seismic loads and the natureof building codes make it necessary to accept someyielding during large earthquakes.3 The design provi-sions in modern seismic codes could not create a purelyelastic response during a large earthquake; in any case,building a structure with such a response would not beeconomical.

Displacement ductility (or just ductility) is the capabil-ity of a structural member or building to distort andyield without collapsing. During an earthquake, a duc-tile structure can dissipate large amounts of seismicenergy after local yielding of connections, joints, andother members has begun.

The actual ductility of a joint or structural member isspecified by its ductility factor, �. There are a number ofdefinitions of the ductility factor, all of which representthe ratio of some property at failure (i.e., fracture) tothat same property at yielding. For example, the ductil-ity factor may be specified in terms of energy absorp-tion, as in Eq. 5.1.

� ¼ U fracture

U yield5 :1

In addition to the definition based on the ratio of ener-gies, there are definitions of the ductility factor basedon ratios of linear strain and angular strain (rotation).These definitions are not interchangeable, althoughthey are related.4 Generally, however, the basic concept

Table 5.1 Scale Factors for Other Recurrence Intervals (based onelastic response to the 1940 El Centro earthquake)

recurrenceinterval (yr)

scalefactor

2 2.7720 1.8332 1.5070 1.00

Figure 5.3 Two Types of Log Tripartite Plots

log Sd

log S a

log S d

log f

log

Sv

log Sa

log T

log

Sv

(a) (b)

3The high seismic loading expected in California and the high cost of atotally elastic design make it necessary to accept some yielding. There-fore, the building is designed to withstand a smaller effective peakacceleration (see Sec. 3.5) without yielding, thereby ensuring yieldingwhen a larger ground acceleration is experienced.4For ideal (linear) elastoplastic systems, the ductility based on energyabsorption, �U, can be calculated from the ductility based on strain,��, as

�U ¼ 2�� � 1

This means that if the ductility, as calculated from linear strain, is4 to 6, the ductility will be 7 to 11 when calculated from Eq. 5.1.

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R E S P O N S E O F S T R U C T U R E S 5-3

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(i.e., the ratio of some failure property to the same yieldproperty) is all that is needed to explain the significanceof a ductile structure.

The minimum assumed ductility (based on strain ordeformation) of building structures with good connec-tions and good redundancy that are designed to modernseismic codes is 2.2. (Ductility of bridge structures ismuch less.) Desirable levels vary, although it is best tohave large values of the ductility factor—4 to 6 forconcrete frames and 6 to 8.5 for steel frames. In orderto achieve these levels of ductility in the structure over-all, the structural members themselves must have spe-cial detailing with inelastic deformation in mind.

6. STRAIN ENERGY AND DUCTILITYFACTOR

The area under the stress-strain curve represents thestrain energy absorbed, U, as shown by Fig. 5.5. Themaximum energy that can be absorbed without yielding

(i.e., the area under the curve up to the yield point) isknown as the modulus of resilience, UR. The maximumenergy that can be absorbed without failure is the mod-ulus of toughness (rupture), UT. One definition of theductility factor, �, can be calculated from the ratio ofthese two quantities.

� ¼ UT

UR5 :2

Figure 5.5 Strain Energy

+yield

σ

ϵ

rupture

UR

UT

Figure 5.4 Elastic Log Tripartite Plot (1940 El Centro earthquake)

0.01 g

0.02 g

0.03 g0.0

4 g0.05 g

0.1 g

0.2 g

0.3 g0.4

g0.5 g

1 g

2 g3 g

4 g5 g

10 g

20 g

0.02″

0.03″0.04″

0.05″

0.1″

0.2″

0.3″

0.5″0.4″

1″

2″

3″

5″

20″10″

100″

30″

40″

50″

4″

spec

tral d

isplac

emen

t S, in

spectral acceleration Sa , gravitiesupper

bound o

f gro

und motio

n (idea

lized

)

ζ=0

ζ=0.02

ζ=0.20

ζ=0.10

0.051

2

3

4

5

10

20

30

40

50

80

0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 10undamped natural period T (sec)

spec

tral

vel

oci

ty S

v (i

n/s

ec)

Reprinted from Design of Multistory Reinforced Concrete Buildings for Earthquake Motions, by John A. Blume, Nathan M. Newmark, and Leo H. Corning, 1961, with permission from the Portland Cement Association, Skokie, IL.

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