Comparison of Drift Control Criteria as Nominated by International Seismic Design Standards

Luis Andrade

1

1Senior Structural Engineer, Prisma Ingeniería, Peru

Synopsis: Major seismic events around the world have displayed the close relationship between lateral displacements, and structural and non-structural damage in buildings. Whilst most seismic design standards are presently force-based, the majority have established limits on lateral drift. This paper documents a study of the stiffness characteristics nominated in several seismic codes, focusing in particular on the requirement to control lateral drift. A method for evaluating and comparing the control of inter-storey displacements is proposed, and an evaluation of seismic codes is undertaken, including: Australia (1), Chile (2), Colombia (3), Europe (4), New Zealand (5), Panama (6), Peru (7), Turkey (8) and the USA (9, 10 & 11). The results of this study show that the Chilean code is the most stringent in controlling lateral displacements. In the short period region (up to 0.13 sec) the Colombian, Peruvian and New Zealand codes are among the most stringent, while for periods in between 0.13 sec to 2.85 sec the Eurocode 8 is second only to the Chilean. For longer periods, the Colombian (2.85 sec), Peruvian (3.70 sec), Panamanian (4.00 sec), Turkish (4.75 sec), American (4.90 sec and 5.30 sec), New Zealand (7.70 sec), and Australian (9.65 sec) seismic standards, respectively, are all more stringent than the European normative. Among the least rigorous standards is the Panamanian for periods up to 2.65 sec, with the Australian standard the least stringent for periods up to 9.65 seconds. A direct comparison of the major seismic codes of the USA and Europe shows that the latter are more rigorous up to a period of 4.9 sec. The opposite applies thereafter, with American standards more stringent for longer period structures. Keywords: inter-storey drift, seismic codes, lateral displacement, lateral drift limits, acceleration response spectrum, displacement response spectrum, earthquake.

1. Introduction

The philosophy of aseismic design in most seismic standards is based on ensuring that a building will not collapse when subject to the most severe earthquake likely to occur during the economic life of a building (unless the structure is alternately fitted with base isolation or passive dissipative systems). Whilst minimising collapse this philosophy deliberately allows for major non-structural damage. Structural elements, specifically beam to column connections and shear walls are permitted to undergo minor (reparable) damage to the extent of developing plastic hinges.

Lateral displacements and inter-storey drift have three primary effects on a structure: damage to structural elements (such as beams, columns and shear walls); damage to non structural elements (such as windows, infill walls, partition walls, false ceiling, cladding, etc); and displacements can also affect adjacent structures. Therefore without proper consideration during the design process, large displacements and inter-storey drifts can have adverse effects on structural elements, non-structural elements, and adjacent structures.

1.1 Inter-storey Drift vs Lateral Displacements

Lateral displacements are the predicted movement of a structure under lateral loads, with reference to the original storey location in the horizontal plane. Inter-storey drift however is defined as the difference of maximum elastic or elastoplastic lateral displacements of any two adjacent floors under factored loads. While the ‘inter-storey drift ratio’ is defined as the inter-storey drift divided by the respective storey height.

1.2 Seismic Code Requirements for Inter-storey Drifts

Drift control requirements are nominated by the design provisions of most building codes. However, design parameters and the analytical assumptions relating to lateral forces used to calculate drifts vary from code to code. The limiting inter-storey drift values recommended also vary widely, as can be seen in Table 1. It should be noted that most standards apply limiting values to the elastoplastic lateral deflections, however some codes like the Chilean and Turkish, impose limits corresponding to the elastic response of the structure. To enable direct comparison of elastoplastic lateral deflections in Table 1, elastic inter-storey drift limits have been multiplied by their respective lateral force reduction factor. Table 1 includes the main design parameters required to calculate lateral forces for a reinforced concrete building with dual system (a combination of ductile shear walls and moment resisting frames, in which the frames alone are capable of resisting 25% of the lateral shear forces).

Table 1. Main parameters for calculating displacements in different Codes (for RC

structures with Ductile Dual System)

Country/Continent Code

Inertia Required to calculate Lateral

Stiffness in Reinforced Concrete Buildings

Lateral Force Reduction Factor

Lateral Displacement Amplification

Factor(1)

Lateral Displacement Amplification Factor to Lateral Force Reduction

Factor Ratio, F

Drift Ratio Limit

Australia AS

1170.4:2007 Uncracked / Cracked µ / Sp = 5.97 µ / Sp = 5.97 (µ / Sp ) / (µ / Sp ) =1.00 0.0150

Chile NCh433 Uncracked R* varies with

T R* varies with T R* / R* = 1.00 0.002 R*

Colombia NSR-98 Uncracked R varies with

T R varies with T R / R = 1.00 0.010

Europe EC 8 - EN

1998-1:2004 Cracked q = 5.85 q = 5.85 q / q = 1.00 0.010

New Zealand NZ

1170.5:200 Cracked µ / Sp = 8.57

µ kdm varies from 7.20 to 9.00

(µ kdm ) / (µ / Sp ) varies from 0.84 to 1.05

0.025

Panama REP2004 Uncracked R = 8.00 Cd = 6.50 Cd / R = 0.81 0.020

Peru NTE E-030 Uncracked R = 7.00 0.75 R = 5.25 0.75 R / R = 0.75 0.007

Turkey 1997 Uncracked R varies with

T R varies with T R / R = 1.00

0.020 or 0.0035 R

USA

UBC 1997 Cracked R = 8.50 0.70 R = 5.95 0.70 R / R = 0.70 0.020

ASCE 7-05 IBC 2009

Cracked R = 7.00 Cd / I = 5.50 (Cd / I ) / R = 0.79 0.020

(1) elastoplastic to elastic displacement ratio

For the same peak ground acceleration, soil conditions, and structural system, different values for design base shear and lateral deflections are obtained depending on the code adopted. The differences are a combination of the following aspects:

• Cracked or Uncracked Inertia: For reinforced concrete structures, some standards require the adoption of cracked sections in the modelling of a building, while some others allow the use of gross sections (see Table 1). This affects the stiffness of the building, and consequently its fundamental period. A building modelled with uncracked sections has a lower fundamental period and could have a different base shear value than its pair modelled with cracked sections, depending on the shape of the acceleration response spectrum given by the design provisions (usually a larger fundamental period is related to a lower base shear if the building is located in rock or stiff soil). The stiffness of the building not only affects the calculation of inertial forces but also the lateral displacements. The stiffer the building, the lesser are the values of its lateral deflections. There is also disagreement among the standards on the level of cracking that should be adopted for modelling as can be extracted from Table 2.

Table 2. Criteria for adoption of cracked inertias for modelling seismic design

Country/Continent Code Inertia Required to calculate Lateral Stiffness in Reinforced Concrete Buildings

Australia AS

1170.4:2007

(2) In the calculation of deformations and action effects in a structure, for both the serviceability and strength

limit states, an estimate of the stiffness of each member shall be based on either

(a) the dimensions of the uncracked (gross) cross-sections; or

(b) other reasonable assumptions, which better represent conditions at the limit state being considered, provided they are applied consistently throughout the analysis.

Europe EC 8 - EN

1998-1:2004 All structural elements 0.50 Igross

(3)

New Zealand NZ

1170.5:200

The stiffness to be used in the analysis for seismic actions in the ultimate limit state, should be based on the member stiffness determined from the load and deflection that is sustained by the member when either:

(a) the material sustains first yield; or

(b) the material sustains significant inelastic deformation.

USA

UBC 1997 Columns (4)

0.70 Igross

Walls cracked (4)

0.35 Igross

Walls uncracked (4)

0.70 Igross

Beams (4)

0.35 Igross

Flat Plates (4)

0.25 Igross

ASCE 7-05 IBC 2009

(2) extracted from AS-3600:2009 (12)

(3) Igross refers to the uncracked first moment of area of the section of a reinforced concrete element.

(4) extracted from ACI 318-08 (13)

• Lateral Force Reduction Factor: Lateral displacements, are also closely related to the amount of lateral force introduced in a structure during an earthquake. Unfortunately, codes are not in agreement when introducing lateral force reduction factors for calculating design base shears (see Table 1). In the codes of different countries, we can find that for a given material and type of structural system, the amount of elastic force reduction varies considerably from standard to standard. This is indeed another direct cause of the differences in the values of the lateral displacements.

• Fundamental Period Limits: Some standards, like the Americans ASCE 7-05 and IBC 2009, and the Panamanian REP 2004, apply an upper limit for the fundamental period of a structure in order to establish the minimum base shear to be adopted in the design. This greatly influences the amount of lateral force introduced in the design of long fundamental period structures. For a structure with long fundamental period and located in a site on rock or stiff soil, limiting the value of its fundamental period to a lower value means that the building will be designed for a higher base shear, and consequently, higher values for its inter-storey drifts can be expected. Nevertheless, ASCE 7-05 and IBC 2009 permit the calculation of lateral displacements using seismic design forces based on computed fundamental period without considering the upper limit (Cl. 12.8.6.2 from ASCE 7-05). On the other hand, the Panamanian standard does not provide such concession.

• Base Shear Limits: Most standards specify a minimum level of base shear to be applied in the design of buildings; however, the majority of codes permit the calculation of drifts without taking into account this requirement. An exception to the rule are the codes of Colombia, Chile, Panama and Turkey.

• Shape of the Acceleration and Displacement Response Spectrum: The difference in shape of the acceleration response spectrum (Figure 1) and intrinsically, the displacement response spectrum (Figure 2) amongst codes, also has a significant influence in the resulting lateral displacements. Figure 2 shows the displacement response spectrums, as gained from the acceleration response spectrums using Eq. 9 (see Section 2. Code Stringency Index (CSI) Method for Comparing Inter-storey Drift obtained from Spectral Curves). Acceleration spectra as nominated by the various codes tend to be inaccurate (and often excessively conservative) in the long-period range. A more representative alternative to using acceleration spectra to generate displacement spectra, is to use source mechanics, and recent digital records (e.g. Bommer, 2001, Faccioli et al, 2002) (14).

Figure 1. Elastic Acceleration response spectrum for different codes.

Figure 2. Displacement response spectrum for different codes.

Figure 2 displays the variation of displacement spectra as obtained by using the acceleration spectra of various codes. Australian, European, New Zealand and the American ASCE 7-05, all have a branch of constant displacement for their response spectrum in the long period region, however, these branches initiate at different fundamental periods varying from 1.5 sec to 8 sec. For the Chilean, Colombian, Peruvian, Turkish and the American UBC 1997, the displacements exponentially increase, as the fundamental period of the structure increases. The displacement response spectrum as calculated using the Panamanian standard is unique in having an abrupt increase at the fundamental period of 4 sec. All of the above-mentioned standards violate the most basic principle of structural dynamics, as they nominate infinite maximum ground displacements for very long periods (14). All of these codes, in the authors’ opinions, are too conservative because empirical data dictates that displacement spectrums should increase linearly to a corner period, then remain

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Period, T (sec)

Elastic Spectral Acceleration, Sa (g)

Australia - AS 1170.4:2007

Chile - NCh433

Colombia - NSR-98

Eurocode 8 - EN 1998-1:2004

New Zealand - NZ 1170.5:2004

Panama - REP2004

Peru - NTE E-030

Turkey - 1997

USA - UBC 1997

USA - ASCE 7-05/IBC 2009

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Period, T (sec)

Spectral Displacement, Sd (mm) .

Australia - AS 1170.4:2007

Chile - NCh433

Colombia - NSR-98

Eurocode 8 - EN 1998-1:2004

New Zealand - NZ 1170.5:2004

Panama - REP2004

Peru - NTE E-030

Turkey - 1997

USA - UBC 1997

USA - ASCE 7-05/IBC 2009

constant, for large earthquakes, or decrease for moderate earthquakes. At very large periods (eg T>10sec) the response spectrum must decrease to match the peak ground displacement (15). It should also be noted that the fundamental period that defines the transition from increasing to constant spectral displacement levels depends on the type of fault that defines the seismicity of the site, the magnitude and the distance to the fault (16). Concern has been expressed recently that the EC8 spectral ordinates may be excessively low at longer periods (16), particularly if compared with those defined for the USA in ASCE 7-05 and IBC 2009. In the latter codes, the constant displacement plateau begins at periods ranging from 4 to 16 sec, whereas the EC8 Type 1 spectrum (for the higher seismicity regions of Europe) has a constant displacement plateau commencing at just 2 sec.

2. Code Stringency Index (CSI) Method for Comparing Inter-storey Drift obtained from Spectral Curves

2.1 Inter-storey Drift Ratio

For each separate mode of a structure, the maximum response can be obtained directly from the displacement response spectrum (17). For example, the maximum displacement vector in mode ‘n’ is given by:

nnN*

*

max )T,(SDM

Ld φ⋅ξ⋅= (1)

where

*L : participation factor of the system (representing the extent to which the earthquake motion tends to excite the response)

*M : generalized mass of the system

** ML : modal mass participation ratio

)T,(SD nNξ : spectral displacement corresponding to the damping, ξN, and period, TN, of the nth

mode of vibration

nφ : shape vector

A factor F (a ratio between the displacement amplification factor and force reduction factor) is introduced to calculate elastoplastic displacements by amplifying elastic displacements. For example, because elastoplastic displacements are calculated from elastic displacements when using the American ASCE 7-05 / IBC 2009 and UBC 1997, these codes nominate an F factor of Cd/IR and 0.7 respectively. For Eurocode 8 the F factor is 1, as its ratio for the elastoplastic to elastic lateral displacements is q (see Table 1 showing a list of F factors for other codes). Introducing F in Eq. 1, the maximum displacement vector is given by:

F)T,(SDM

Ld nnN*

*

max ⋅φ⋅ξ⋅= (2)

The inter-storey drift ratio, max∆ , can now be obtained dividing Equation 2 by the inter-storey

height, h:

F)T,(SDM

L

h

1

h

dnnN*

*max

max ⋅φ⋅ξ⋅⋅==∆ (3)

From Equation 3 we can easily obtain the maximum inter-storey drift ratio of a given level,

max∆ , directly from the displacement response spectrum, and express its value in terms of a

magnitude (in lieu of a vector) as:

F)T,(SDM

L

h

1

h

dnnN*

*max

max ⋅φ⋅ξ⋅⋅==∆ (4)

2.2 Drift Control Index

The maximum drift ratio obtained in the analysis (max∆ ) must be less than the upper limit (

itlim∆

) stipulated by the code/s to ensure the structure under interrogation satisfies code drift limits. The relationship between the maximum drift ratio obtained from analysis, and the drift ratio limit given by the standards, should thus be less than 100%. To this end, the author proposes a Drift Control Index (i), given by:

(%)iitlim

max

∆

∆= (5)

2.3 Code Stringency Index

To compare how stringent a code is in relation to another, the author proposes a factor nominated as the Code Stringency Index, or CSI, defined as:

(%)i

iCSI

CodeY

CodeX= (6)

Incorporating Eq. 5 in Eq. 6 we obtain the following expression for the CSI:

(%)/F)T,(SD

/F)T,(SDCSI

CodeYitlimCodeYCodeYnn

CodeXitlimCodeXCodeXnn

∆⋅ξ

∆⋅ξ= (7)

Subscripts Code X and Code Y, indicate that we are comparing Code X against Code Y in the above formula. For example, if the CSI is lower than 100%, it means that Code Y is more stringent than Code X, while if CSI is higher than 100% the opposite applies. Having a CSI equal to 100% means that both Codes X and Y are equally stringent.

The CSI can also be expressed in terms of the spectral acceleration, )T,(SA nNξ , and the

structure’s lateral stiffness, K, as we know that (17):

M

K=ω (8)

and

)T,(SAK

M)T,(SA

1)T,(SD NNNN2NN ξ⋅=ξ⋅

ω=ξ (9)

where

ω : circular frequency of vibration of the equivalent SDOF (Hz)

M : mass of the building

K : lateral stiffness of the building

)T,(SA nNξ : spectral acceleration corresponding to the damping, ξN, and period, TN, of the nth

mode of vibration

As discussed in Section 1 of this paper, some standards require calculating lateral displacements in concrete structures by considering cracked sections, while others allow calculating displacements with gross sections. To account for this in our stringency comparison we must introduce lateral stiffness into the equation for CSI. Introducing Eq. 9 in Eq. 7, we obtain:

[ ] [ ][ ] [ ]

(%)K/F)T,(SA

K/F)T,(SACSI

CodeYCodeYitlimCodeYCodeYnn

CodeXCodeXitlimCodeXCodeXnn

⋅∆⋅ξ

⋅∆⋅ξ= (10)

where

CodeXK ,CodeYK : type of lateral stiffness (cracked or uncracked) stipulated in Code X or Code Y,

respectively.

3. Code Stringency Index Comparison from Spectral Curves

The CSI equation proposed in Eq. 10 enables comparison of the code’s stringency with respect to limiting lateral displacements for a series of structures with different fundamental periods. Consider a set of buildings for which fundamental periods vary from close to 0 sec to up to 10 sec, and which are situated on very stiff soil with 0.4g peak ground acceleration (10% probability of being exceeded in 50 years, or 475 years mean return period). Assume that all buildings are made of reinforced concrete, and consist of dual structural systems incorporating both shear walls and moment resisting frames. For those codes which require the modeling of buildings with cracked inertia, assume a lateral stiffness which is half the lateral stiffness of the same structures with uncracked sections (as recommended in the Eurocode 8). Some other parameters of importance are taken as per Table 1. Figure 4 shows the results of this comparison against the Eurocode 8 (i.e. parameters of the European standard are used as the denominator in Eq. 10):

Figure 4. CSI values for various codes.

The results displayed by Figure 4 show that the Chilean standard is the most stringent in controlling lateral displacements for the whole spectrum of fundamental periods. For very short periods, up to 0.13 sec, Eurocode 8 is less stringent than the Colombian, Peruvian and New Zealand design provisions, but is the second most stringent for periods between 0.13 sec to 2.85 sec. The Panamanian code is the least stringent up to a period of 2.65, the Australian code is least rigorous for periods between 2.65 and 9.65 sec, after which Eurocode 8 becomes the least stringent in requiring lateral displacement control. Despite being the less stringent in the short fundamental periods range, the Panamanian standard is the third more rigorous for periods larger than 4 sec after the Chilean and Colombian standards. A comparison among the American and the European design provisions (which are the codes of choice in countries where there is a lack of seismic normative) indicates that Eurocode 8 is more stringent for periods of up to 4.9 sec, after which the American ASCE 7-05/IBC 2009 and UBC 1997 take over.

10

100

1000

10000

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Period, T (sec)

CSI (%)

Australia - AS 1170.4:2007

Chile - NCh433

Colombia - NSR-98

Eurocode 8 - EN 1998-1:2004

New Zealand - NZ 1170.5:2004

Panama - REP2004

Peru - NTE E-030

Turkey - 1997

USA - UBC 1997

USA - ASCE 7-05/IBC 2009

4. Conclusions & Recommendations

In studying and comparing seismic design provisions of various codes it is evident that lack of uniformity exists regarding calculation of lateral displacements. Differences result from varying factors including drift limit ratios, the use of cracked or uncracked sections, lateral force reduction factors, upper limits to the fundamental period of vibration, minimum base shear to be considered, and the shape of acceleration and displacement response spectrums. The author’s proposes that further consideration should be given in defining the displacement response spectrum, and support the changing trend in seismic design philosophy from force-based towards a displacement-based. Force-based codes of practice sometimes lead the engineer towards expensive designs and unfeasible structures. A code stringency index, CSI, is proposed to enable comparison of code stringency in

controlling lateral displacements obtained from spectral acceleration curves. The comparison of several international standards shows that the Chilean code is the most stringent standard over the whole range of periods (up to 10 sec). This probably explains why most structures designed with the current Chilean seismic normative have had a good performance during the 8.8 magnitude earthquake that rocked the central and southern region of the South American country on the 27

th February 2010.

For short periods (up to 0.13 sec) the Colombian, Peruvian and New Zealand codes rank among the most stringent, while, for periods in between 0.13 sec to 2.85 sec Eurocode 8 is second to the Chilean. For longer periods the Colombian (2.85 sec), Peruvian (3.70 sec), Panamanian (4.00 sec), Turkish (4.75 sec), American (4.90 sec and 5.30 sec), New Zealand (7.70 sec) and Australian (9.65 sec) seismic standards, are all more stringent than the European normative. Among the least rigorous standards is the Panamanian for periods up to 2.65 sec, after which the Australian becomes the least stringent up to a period of 9.65 seconds. This makes the Australian standard one of the least conservative for the design of long period structures (particularly high-rise buildings). A direct comparison of the seismic codes of the USA and Europe showed that the latter are more rigorous up to a period of 4.9 sec. The opposite applies thereafter, were for longer period structures the American standards are more stringent. In view of the above, it seems the American UBC is most applicable to regions such as Dubai, were high rises like the Burj Dubai have reported fundamental periods up to 11.3 sec (18). It is recommended that the displacement response spectrums nominated by Eurocode 8 should be revised, as it is found that its ordinates are exceedingly low for longer periods, especially in comparison to ASCE 7-05 and IBC 2009.

5. Acknowledgments

The author expresses his sincere gratitude to Alejandro Muñoz, Professor of the Pontifical Catholic University of Peru, for his constructive critique provided during the production of this paper.

6. References

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