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Earthquake Loads According to IBC 2003

The process of determining earthquake loads according to IBC 2003 Spectral Design Method can be broken down into the following basic steps:

• Determination of the maximum considered earthquake and design spectral response accelerations.

• Determination of the seismic base shear associated with the building or the

structure’s fundamental period of vibration. • Distribution of the seismic base shear within the building or the structure.

IBC Safety Concept • The IBC intends to design structures for “collapse prevention” in the event of an

earthquake with a 2 % probability of being exceeded in 50 years

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Introduction Seismic Response Spectra: - A response spectrum provides the maximum response of a SDOF system, for a given damping ratio and a range of periods, for a specific earthquake. - A design response spectrum is a smoothed spectrum used to calculate the expected seismic response of a structure Figure (1) shows six inverted, damped pendulums, each of which has a different fundamental period of vibration. To derive a point on a response spectrum, one of these pendulum structures is analytically subjected to the vibrations recorded during a particular earthquake. The largest acceleration of this pendulum structure during the entire record of a particular earthquake can be plotted as shown in Figure 1(b). Repeating this for each of the other pendulum structures shown in Figure 1(a) and plotting and connecting the peak values for each of the pendulum structures produces an acceleration response spectrum. Generally, the vertical axis of the spectrum is normalized by expressing the computed accelerations in terms of the acceleration due to gravity g . In Figure (2), displacement, velocity, and acceleration spectra for a given earthquake are shown. In this figure, structures with short periods of 0.2 to 0.5 seconds are almost rigid and are most affected by ground accelerations. Structures with medium periods ranging from 0.5 to 2.5 seconds are affected most by velocities. Structures with long periods greater than 2.5 seconds, such as tall buildings or long span bridges, are most affected by displacements.

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Viscous damping

(a) Damped pendulums of varying natural frequencies

0 1.0 2.0 3.0 4.0

0.5% Damping

2% Damping

5% Damping

4

3

2

1

Acc

eler

atio

n Sa

Natural period of vibration, T (sec)

Acceleration response spectrum

Figure (1): Earthquake Response Spectrum

Reference: MacGregor, J and Wight, J., "Reinforced Concrete Mechanics and Design" 4th Edition, Prentice Hall, NJ, 2005.

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Analysis Procedure

1- Determination of maximum considered earthquake and design spectral response accelerations:

• Determine the mapped maximum considered earthquake MCE spectral response accelerations, sS for short period (0.2 sec.) and 1S for long period (1.0 sec.) using the spectral acceleration maps in IBC Figures 1615(1) through 1615(10). Straight-line interpolation is allowed for sites in between contours or the value of the higher contour shall be used. Acceleration values obtained from the maps are given in % of g , where g is the gravitational acceleration.

• Determine the site class, which is based on the types of soils and their engineering

properties, in accordance with IBC Section 1615.1.1. Site classes A, B, C, D, E, and F, obtained from Table 1615.1.1, are based on the average shear velocity, sv , average standard penetration resistance, N , or the average undrained shear strength, us . These parameters represent average values for the top 30 m of soil. When the soil properties are not known in sufficient detail to determine the site class, site class D shall be used. Unless the building official determines that the site class E or F is likely to be present at the site.

• Determine the maximum considered earthquake spectral response accelerations

adjusted for site class effects, MSS at short period and 1MS at long period in accordance with IBC 1615.1.2.

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138

saMS SFS = 11 SFS vM =

where:

aF = short-period site coefficient, given in Table 1615.1.2(1) vF = long-period site coefficient, given in Table 1615.1.2(2)

• Determine the 5% damped design spectral response accelerations DSS at short

period and 1DS at long period in accordance with IBC 1615.3.

MSDs SS )3/2(= 11 )3/2( MD SS =

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2- Determination of seismic use group and occupancy important factor:

• Each structure shall be assigned a seismic use group and a corresponding occupancy importance factor EI , in accordance with Table 1604.5. Seismic use group I are structures not assigned to either seismic use group II or III. Seismic use group II are structures the failure of which would result in a substantial public hazard due to occupancy or use as indicated in Table 1604.5. Seismic use group III are structures required for post earthquake recovery and those containing substantial quantities of hazardous substances as indicated in Table 1604.5.

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3- Determination of seismic design category: All structures shall be assigned to a seismic design category based on the seismic use group and the design spectral response acceleration coefficients, DSS and 1DS . Each building and structure shall be assigned to the worst severe seismic design category in accordance with Table 1616.3(1) or 1616.3(2), irrespective of the fundamental period of vibration of the structure.

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4- Determination of the Seismic Base Shear: 4-1 Simplified Analysis:

• A simplified analysis, in accordance with Section 1617.5, shall be determined to be used for any structure in Seismic Use Group I, subject to the following limitations, or a more rigorous analysis shall be made:

1- Building of light-framed construction not exceeding three stories in height,

excluding basement.

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2- Building of any construction other than the light-framed construction, not exceeding two stories in height, excluding basement, with flexible diaphragm at every level.

• Since the above limitations rule out the use of this method for concrete buildings, it will not be covered here.

4-2 Index Force Analysis: Structures assigned to Seismic Design Category A need only comply with the requirements of Section 1616.4.1 through 1616.4.5, summarized below:

• Structures shall be provided with a complete lateral force resisting system designed to resist the minimum lateral force, xF , applied simultaneously at each floor level according to the following equation:

xx wF 01.0=

Where: xF = The design lateral force applied at level x

=xw The portion of the total gravity load of the structure, W , located or assigned to level x

=W The total dead load and other loads listed below: 1- In areas used for storage, a minimum of 25 % of the reduced floor live load. 2- Where an allowance for partition load is reduced in the floor load design, the

actual partition weight or 2/50 mkg of the floor area, whichever is greater.

3- The total weight of permanent equipment. 4- 20 % of flat roof snow load where flat roof snow load exceeds

2/145 mkg . • The direction of application of seismic forces used in design shall be such that

which will produce the most critical load effect in each component. • The design seismic forces are permitted to be applied separately in each of the

two orthogonal directions. • Load combinations as per Section 9.2 of ACI Code.

4-3 Equivalent Lateral Force Analysis: Section 9.5.5 of ASCE 7-02** shall be used. **ASCE, ASCE Standard Minimum Design Loads for Buildings and Other Structures, ASCE 7-02, American Society of Civil Engineers, Reston, VA, 2002.

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• The seismic base shear V in a given direction is determined in accordance with the following equation:

WCV s=

where:

sC = Seismic response coefficient

( ) ( )TIRS

IRS

E

D

E

DS

//1≤=

DSS044.0≥ R = Response modification coefficient, given in Table 1617.6.2

EI = Seismic occupancy importance factor T = Fundamental period of vibration An approximate value of aT may be obtained from:

75.0nTa hCT =

where: TC = Building period coefficient

= 0.073 for moment frames resisting 100% of the required seismic force = 0.049 for all other buildings

nh = Height of the building above the base in meters The calculated fundamental period, ,T cannot exceed the product of the coefficient, uC , in the following table times the approximate fundamental period, aT . The base shear V is to be based on a fundamental period, T , in seconds, of 1.2 times the coefficient for the upper limit on the calculated values, uC , taken from the following table, times the approximate fundamental period, aT

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Vertical Structural Irregularities Irregularity Type and Description 1a- Stiffness Irregularity- Soft Story A soft story is one in which the lateral stiffness is less than 70 percent of that in the story above or less than 80 percent of the average stiffness of the three stories above. 1b- Stiffness Irregularity- Extreme Soft Story An extreme soft story is one in which the lateral stiffness is less than 60 percent of that in the story above or less than 70 percent of the average stiffness of the three stories above. 2- Weight (Mass) Irregularity Mass irregularity shall be considered to exist where the effective mass of any story is more than 150 percent of the effective mass of an adjacent story. A roof that is lighter than the floor below need not be considered. 3- Vertical Geometric Irregularity Vertical geometric irregularity shall be considered to exist where the horizontal dimension of the lateral force-resisting system in any story is more than 130 percent of that in an adjacent story. 4- In-Plane Discontinuity in Vertical Lateral Force-Resisting Elements An in plane offset of the lateral load-resisting elements greater than the length of these elements or a reduction in stiffness of the resisting element in the story below. 5- Discontinuity in Capacity-Weak Story A weak story is one in which the story strength is less than 80 percent of that in the story above. The story strength is the total strength the story above or less than 80 percent of that in the story above. The story strength is the total strength of all seismic-resisting elements sharing the story shear for the direction under consideration.

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Plan Structural Irregularities Irregularity Type and Description 1a- Torsional Irregularity – to be considered when diaphragms are not flexible Torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at one end of the structure transverse to an axis is more than 1.2 times the average of the story drifts at the two ends of the structure. 1b- Extreme Torsional Irregularity – to be considered when diaphragms are not flexible Extreme torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at one end of the structure transverse to an axis is more than 1.4 times the average of the story drifts at the two ends of the structure. 2- Re-entrant Corners Plan configurations of a structure and its lateral force-resisting system contain re-entrant corners, where both projections of the structure beyond a reentrant corner are greater than 15 % of the plan dimension of the structure in the given direction. 3- Diaphragm Discontinuity Diaphragms with abrupt discontinuities or variations in stiffness, including those having cutout or open areas greater than 50 % of the gross enclosed area of the diaphragm, or changes in effective diaphragm stiffness of more than 50 % from one story to the next. 4- Out-of-plane Offsets Discontinuities in a lateral force path, such as out-of-plane offsets of the vertical elements. 5- Nonparallel Systems The vertical lateral load-resisting elements are not parallel to or symmetric about the major orthogonal axes of the lateral force-resisting system.

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147

148

149

150

Coefficient for Upper Limit on Calculated Period Design Spectral Response, 1DS Coefficient uC

4.0≥ 0.3 0.2

0.15 1.0≤

1.2 1.3 1.4 1.5 1.7

In cases where moment resisting frames do not exceed twelve stories in height and having a minimum story height of 3 m, an approximate period aT in seconds in the following form can be used: NTa 1.0= where N = number of stories

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5- Vertical Distribution of Forces: The vertical distribution of seismic forces is determined from:

VCF vxx = and

∑=

= n

ii

ki

kxx

vx

hw

hwC

1

where: xF = Lateral force at level x vxC = Vertical distribution factor

V = total design lateral force or shear at the base of the building xw and iw = the portions of W assigned to levels xand i

xh and ih = heights to levels xand i k = a distribution exponent related to the building period as follows: k = 1 for buildings with T less than or equal to 0.5 seconds k = 2 for buildings with T more than or equal to 2.5 seconds Interpolate between k = 1 and k = 2 for buildings with T between 0.5 and 2.5

6- Horizontal Distribution of Forces and Torsion: Horizontally distribute the shear xV

∑=

=x

iix FV

1

where: iF = Portion of the seismic base shear, V , introduced at level i

Accidental Torsion, taM

taM = ( )BVx 05.0 Total Torsion, TM tatT MMM +=

F

F

F wn

wx

w1

h

hh

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7- Overturning Moments: The overturning moment xM is given by the following equation:

( )xi

n

xiix hhFM −= ∑

=

τ

where:

iF = Portion of the seismic base shear, V , introduced at level i =τ Overturning moment reduction factor

= 1.0 for the top 10 stories = 0.8 from the 20th story from the top and below = Values between 1.0 and 0.8 determined by a straight linear interpolation for

stories between the 20th and 10th stories below the top

8- Story Drift: The story drift, ∆ , is defined as the difference between the deflection of the center of mass at the top and bottom of the story being considered.

E

xedx I

C δδ =

Where:

dC = Deflection amplification factor, given in Table 1617.6.2 xeδ = Deflection determined by elastic analysis

The allowable story drifts, ∆ , are shown in Table 1617.3.1. 9- P-delta Effect: The P-delta effects can be ignored if the stability coefficient, θ , from the following expression is equal to or less than 0.10.

25.05.0≤≤

∆=

βθ

ddsxx

x

CChVP

Where:

xP = Total unfactored vertical design load at and above level x

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xV = Seismic shear force acting between level x and 1−x sxh = Story height below level x

∆ = Design story drift occurring simultaneously with xV β = Ratio of shear demand to shear capacity for the story between level x and

1−x . Where the ratio β is not calculated, a value of β = 1.0 shall be used.

When the stability coefficient, θ , is greater than 0.10 but less than or equal to maxθ , P-delta effects are to be considered. To obtain the story drift for including the P-delta effects, the design story drift shall be multiplied by )1/(0.1 θ− . When θ is greater than maxθ , the structure is potentially unstable and has to be redesigned. 10- Combination of Load Effects: The value of seismic load E for use in ACI 318-08 load combinations is defined by the following equations for load combinations in which the effects of gravity loads and seismic loads are additive:

DSQE DSE 2.0+= ρ DSQE DSE 2.0+Ω= o (Need not apply to SDC A)

where: E = the effect of horizontal and vertical earthquake-induced forces

DSS = the design spectral response acceleration at short period D = the effect of dead load ρ = the reliability factor related to the extent of structural redundancy of the lateral

force resisting system EQ = the effect of horizontal seismic forces oΩ = the system over strength factor given in Table 1617.6.2.

The value of seismic load E for use in ACI 318-08 load combinations is defined by the following equations for load combinations in which the effects of gravity loads and seismic loads are counteractive:

DSQE DSE 2.0−= ρ

DSQE DSE 2.0−Ω= o (Need not apply to SDC A)

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Redundancy: Seismic Design Categories A, B, and C: For structures in seismic design categories A, B and C, the value of ρ may be taken as 1.0. Seismic Design Category D: For structures in seismic design category D, ρ shall be taken as the largest of the values of xρ calculated at each story of the structure “x” in accordance with this equation

xx Ar

xmax

10.62 −=ρ

where: xA = the floor area in square meters of the diaphragm level immediately above the story.

xrmax = the ratio of the design story shear resisted by the single element carrying the

most shear force in the story to the total story shear for a given direction of loading. For moment frames,

xrmax shall be taken as the maximum of the sum of the shears in any

two adjacent columns in the plane of a moment frame divided by the story shear. For columns common to two bays with moment resisting connections on opposite sides at the level under consideration, 70 percent of the shear in that column may be used in the column shear summation. For shear walls,

xrmax shall be taken equal to the maximum ratio, ixr , calculated as the

shear in each wall or wall pier multiplied by 3.3/ wl , where wl is the wall or wall pier length in meters divided by the story shear and where the ratio 3.3/ wl need not be taken greater than 1.0 for buildings of light frame construction. For dual systems,

xrmax shall be taken as the maximum value as defined above

considering all lateral-load-resisting elements in the story. The lateral loads shall be distributed to elements based on their relative rigidities considering the interaction of the dual system. For dual systems, the value of ρ need not exceed 80 percent of the value calculated above. The value of ρ need not exceed 1.5, which is permitted to be used for any structure. The value of ρ shall not be taken as less than 1.0. 11- Diaphragm Forces: Diaphragms are designed to resist design seismic forces determined in accordance with the following equation:

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pxn

xii

n

xii

px ww

FF

∑

∑

=

== ranges from pxEDS wIS)4.02.0( →

Where: iF = The design force applied to level i pxF = The diaphragm design force

iw = The weight tributary to level i pxw = The weight tributary to the diaphragm at level x

12- Seismic Detailing Requirements

• Level of detailing required depends on the level of seismic risk:

- Low Seismic Risk: SDC* A, B - Medium Seismic Risk: SDC C - High Seismic Risk: SDC D, E, F

*SDC= Seismic Design Category

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Example (7): For the building shown in Example (1) and using IBC-03 evaluate the forces at the floor levels perpendicular to axes 1-1, 2-2, 3-3 and 4-4. Note that site class is D, g25.0Ss = and g10.0S1 = . Solution:

• Using Tables 1615.1.2(1) and 1615.1.2(2), short-period site coefficient 60.1Fa = and long-period site coefficient 40.2Fv = .

• Maximum considered earthquake spectral response accelerations adjusted for site class effects are evaluated.

( ) g4.0g25.060.1SFS saMS === and

( ) g24.0g10.040.2SFS 1v1M === • The 5% damped design spectral response accelerations DSS at short period and

1DS at long period in accordance are evaluated.

( ) g267.0g40.032S

32S MSDS ===

( ) g16.0g24.0

32S

32S 1M1D ===

• Occupancy importance factor, 0.1IE = as evaluated from Table 1604.5. • From Table 16136.3(1) and for g267.0SDS = , Seismic Design Category (SDC) is

B. For g16.0S 1D = and using Table 1616.3(2), SDC is C. Therefore, seismic design category (SDC) is “C”.

• For ordinary shear walls and using Table 1617.6.2, response modification coefficient 0.5R = .

• The seismic base shear V in a given direction is determined in accordance with the following equation:

WCV s=

( ) ( ) TI/RS

I/RSC

E

1D

E

DSs ≤=

DSS044.0≥

Approximate period ( ) .sec48.021049.0T 75.0a ==

( ) .sec676.048.0408.1TC au == ( ) K.O.sec676.0.sec576.048.02.1T <==

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<== 0534.0

0.5267.0Cs ( ) ( )267.0044.00555.0

576.0)0.5(16.0

>= O.K

i.e., 0534.0Cs = The seismic base shear

( ) tons89.964.18140534.0V ==

• Vertical distribution of forces:

VCF vxx = and ∑

=

=

n

1ii

ki

kxx

vxhw

hwC

K = 1.038 (from linear interpolation).

Shear forces ∑==

x

1iix FV

Overturning moment ( )xin

xiix hhFM −∑τ=

=,

where 0.1=τ

Vertical Distribution of Forces:

Level iw xh ( ) 038.1xx hw vxC

xF 7 259.2 21 495.09 0.35 34.26 6 259.2 18 361.61 0.26 25.02 5 259.2 15 249.39 0.18 17.26 4 259.2 12 158.26 0.11 10.95 3 259.2 9 88.05 0.06 6.09 2 259.2 6 38.54 0.03 2.67 1 259.2 3 9.38 0.01 0.65 0 ∑ 1814.4 0 1400.32 1.00 96.89