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Chapter 9

Seismic Design of Steel Structures

_________________________________________________________________________________

9.1 S16-09: Clauses Clause 27: Seismic Design (ductility of frames) 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9 27.10 27.11 27.12

General Type D (ductile) moment-resisting frames Type MD (moderately ductile) moment-resisting frames MRFs Type LD (limited-ductility) moment-resisting frames Type MD (moderately ductile) concentrically-braced frames Type LD (limited-ductility) concentrically-braced frames Braced Type D (ductile) eccentrically-braced frames Frames Type D (ductile) buckling-restrained braced frames Type D (ductile) plate walls Type LD (limited-ductility) plate walls Conventional construction Special seismic construction.

Annexes:

Rd 5.0 3.5 2.0 3.0 2.0 4.0 4.0 5.0 2.0 1.5

Ro 1.5 1.5 1.3 1.3 1.3 1.5 1.2 1.6 1.5 1.3

-tbd

-tbd

Annex J: Ductile moment-resisting connections Annex L: Design to prevent brittle fracture

NBCC-2010: Division B: Section 4.1.8. Earthquake Load and Effects CE-321 Class Notes Chapter 3 (2012) Reference publications: FEMA (2000) Recommended seismic design criteria for new steel moment frame buildings, FEMA 350, Federal Emergency Management Agency, Washington, DC. Hamburger, Ronald O., Krawinkler, Helmut, Malley, James O., and Adan, Scott M. (2009). "Seismic design of steel special moment frames: a guide for practicing engineers," NEHRP Seismic Design Technical Brief No. 2 available from: http://www.nehrp.gov/pdf/nistgcr9-917-3.pdf

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9.2 Review of concepts in Earthquake Engineering 9.2.1 Basic definitions Earthquake Engineering deals with applying civil engineering design principles to reduce life and economic losses due to earthquakes, (i.e. to mitigate seismic risk). Seismic risk can be defined as the probability of losses occurring due to earthquakes within the lifetime of a structure. The two main components of seismic risk are: a)

Seismic hazard : This component of risk is determined by nature and cannot be reduced. There are several damaging effects due to earthquakes including: ground shaking, landslides, surface ruptures and liquefaction. The level of seismic hazard can vary from having frequent low intensity earthquakes to having rare severe earthquakes.

b)

Structural vulnerability : This component depends on the structural configuration and properties and thus can be reduced by proper seismic design of structures. Steel structures have several inherent characteristics that are advantageous for seismic design. At the top of the list is the high ductility of steel compared to other construction materials. Ductility is the ability of the structure to deform past yielding without significant strength deterioration. However, to make use of the advantages of steel as a construction material for seismic design, the design engineer has to be familiar with the code design and construction provisions. In essence, code provisions are set to avoid different sources of structural vulnerability which include: • inappropriate detailing • inappropriate design • poor connections • irregularities in structural configuration (in plan and/or in elevation) • soft storey (laterally) • pounding against nearby structures • failure to conform to the intent of the design.

9.2.2 Nature of Earthquakes There are several causes for earthquakes: Some are caused by volcanoes which may be a triggering factor for earthquakes, or there can be induced seismicity resulting from underground explosions. However, a cause which is believed to be the main reason for most earthquakes is referred to as “plate tectonics”. Compared to the radius of the earth, the thickness of the earth’s crust is relatively thin. The earth’s crust is composed of several tectonic plates which move relative to each other about 50 mm per year.

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The roughness of the surfaces and edges of tectonic plates, and the huge pressures involved, cause potential sliding and slipping movements to generate friction forces large enough to lock-up surfaces in contact. Instead of sliding past each other, rock in a plate boundary area absorbs greater and greater compression and shear strains until it suddenly ruptures. At rupture, the accumulated energy (strain energy) within the strained rock mass releases in a sudden manner with a violent jarring motion. This is an earthquake.

Most earthquakes are caused by movement between tectonic plates: 70% around the perimeter of the pacific plate; 20% along the southern edge of the Eurasian plate and 10% cannot be explained by plate tectonics some of which are intra-plate (within the plate). The surface along which the crust of the earth fractures is an earthquake fault. The point in the fault surface area considered the centre of energy release is termed the focus, its projection up to the earth’s surface defines the epicenter. The distance between the focus and the epicenter is known as the focal depth.

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When the fault ruptures, seismic waves emanate in all directions from the focus. The two types of underground waves which are generated by the fault rupture are: • P waves (Primary waves) also known as “compression waves”, they push and pull the soil through which they pass. • S waves (Shear waves) they move soil particles side-to-side either horizontally or vertically. This “shear effect” is of most concern and damaging to buildings. There is also a surface-rippling wave known as a Rayleigh wave. The response of the soil affects the features of the earthquake waves felt by the buildings. For example: deep layers of soft soil, as may be found in river valleys, significantly amplify shaking and also modify the frequency content of seismic waves by filtering out higher frequency excitations.

9.2.3 Earthquake Magnitude (Richter) and Intensity (Mercalli) Earthquake magnitude defines the amount of energy released by the earthquake. Thus, earthquake magnitude is a quantitative measure of earthquake severity commonly measured by the Richter scale(1935). Each earthquake is assigned only one magnitude value. This is determined by seismologists from seismograph records. The Richter scale is a magnitude scale for earthquakes which relates logarithmically to the amount of energy released. This means that an increase of “1” in the number on the Richter scale represents a ten-fold (101) increase in amplitude and a 30-fold (√ 103) increase in discharged energy.

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On the other hand, earthquake intensity is a qualitative description of the earthquake severity. Accordingly, earthquake intensity varies according to the location where shaking is felt. Factors affecting earthquake intensity at a certain location includes: earthquake magnitude, epicentral distance and soil conditions. The most recognized intensity scale for earthquakes is the Modified Mercalli Intensity Scale summarized in the following Table. Intensity

Description

I to III

Not felt, unless under special circumstances.

IV

Generally felt, but not causing damage.

V

Felt by nearly everyone. Some cracked plaster. Some crockery broken or items overturned.

VI

Felt by all. Some fallen plaster or damaged chimneys. Some heavy furniture moved.

VII

Negligible damage in well designed and constructed buildings through to considerable damage in construction of poor quality. Some chimneys broken.

VIII

Depending on the quality of design and construction, damages ranges from slight through to partial collapse. Chimneys, monuments and walls fall.

IX

Well designed structures damaged and permanently racked. Partial collapses and buildings shifted off their foundations.

X

Some well-built wooden structures destroyed along with most masonry and frame structures.

XI

Few, if any masonry structures remain standing.

XII

Most construction is severely damaged or destroyed.

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9.2.4 Philosophy of Seismic-Resistant Design Given the uncertainties in determining the degree of severity of earthquakes and for obvious economical reasons, it is unrealistic to design a structure to respond elastically during a major earthquake. Thus, the philosophy of seismic design for major earthquakes is “collapse prevention rather than damage prevention”. For less severe earthquakes, lower Performance Based Earthquake levels of damage are accepted. As Engineering Frequent the importance of the structure Minor increases, the criteria of accepted H performance are more stringent as a z shown in the next figures taken from alev SEAOC. rel d drza a

LH e v e l

Performance objective(s)

Performance levels

Very rare Severe Fully operational EQ. Eng.

Risk

Hazard

Collapse Vulnerability

PBEE

Quantification Response Spectrum

Performance Objectives S e i s m i c H as z le v a el rd r daz

ah Lic si em v eS e l s

Performance levels Operational

Immediate occupancy

Life safety

Structural stability

Fully operational

Operational

Life safety

Near Collapse

Frequent (Low Intensity) Occasional

Ra re

Very rare (Severe intensity) Vision 2000 (SEAOC 1995) EQ. Eng.

Risk

Hazard

Vulnerability

PBEE

Quantification Response Spectrum

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9.3 Seismic Forces on Buildings 9.3.1 Nature of Seismic-Induced Forces

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The ground motion caused by earthquakes is experienced by the building as acceleration at the base. The masses of the building resist the ground motion acceleration and inertia forces are generated. The inertia force can be quantified using Newton’s second law of motion F = M × a where M is the mass of the building, and a is the acceleration of the mass caused by ground motion acceleration. Inertia forces are generated throughout the building on every element and component. However, most of the mass of the building is concentrated in its floors and roof. For this reason and for simplicity of design, engineers assume inertia forces act at the center of mass of the roof and floors as “lumped masses”. Noteworthy of comparison at this point are the similarities and differences of seismic to wind-induced forces: Seismic-induced forces Wind-induced forces mainly lateral mainly lateral - some earthquakes have - near-vertical wind-induced significant vertical suction forces acting on roofs of components ( normally have little impact on horizontal motion). the building behavior dynamic dynamic - peak seismic-induced forces - strong wind gusts can last for act for fractions of a second several seconds - act within the building mass - act external to the building

Inertia Forces

Loading

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9.3.2 Factors affecting Seismic-Induced Forces (a) Building weight: When an object is subjected to a dynamic action, the inertia force is proportional to its mass according to Newton’s second law of motion. Thus, as the weight of an object increases, the inertia force increases for a given level of acceleration. Buildings with heavier-weight structural materials are subjected to higher levels of seismic-induced forces than lighter ones. It is therefore advisable to use lighter weight construction in seismic prone areas (less mass gets excited). (b) Natural periods of vibration: A natural period of vibration is the time for one complete vibration cycle that a structure would undertake when subjected to an initial dynamic stimulus and then left to oscillate freely. The lowest frequency has the largest natural time period of vibration and is called the 1st mode or fundamental mode of the structure. Depending on the structural and geometrical configuration of the structure, there may be other periods corresponding to higher-order (2nd, 3rd, ...) modes. It is noteworthy that the contribution of the first (fundamental) mode of vibration is the most prominent and important for low and medium-rise buildings.

(c) Damping: Damping is a resistance to free vibration and defines the energy-dissipation mechanism which steadily diminishes the amplitude of vibration. Damping in structures is mainly caused by internal friction within building elements. The type of construction material affects the degree of damping. There are many forms of damping. Viscous damping is velocity dependent. Currently, the most popular form is “proportional damping” (mass and stiffness dependent); it is also known as: “Rayleigh”, “classical”, “orthogonal” or “modal” damping. For additional information on damping refer to Tedesco et al.(1999).

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9.3.3 *** Response Spectrum concept *** Response spectrum is a powerful “response vs. time” graphical analytical tool used to quantify effects of natural periods of vibration on responses (acceleration, velocity or displacement) of buildings to an earthquake. Design codes use this response spectrum to develop design spectra.

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9.4 Lateral Load Resisting Systems 9.4.1 Lateral Load Resistance The two main types of gravity load resisting systems are: o skeleton type structures: consisting of beams and columns. o wall bearing Structures The vertical members of both systems are mainly subjected to compression forces with the requirement to have sufficient cross section to resist buckling. The instability under lateral forces is a main issue for both systems.

A main principle of seismic-resistant design is to ensure collapse prevention; therefore it is essential to design lateral-load resisting systems with lateral stability. NBCC-2010 and S16-09 seismic design provisions for lateral load resistance are based on the “capacity design procedure”. In this design procedure, certain structural components are designed to act as structural fuses (sacrificial elements). Specifically designed and detailed, these components are to fail and exhibit inelastic response dissipating energy during a design level earthquake. The locations of these components are engineered such that the gravity load-carrying capacity of the whole system is not impaired due to the damage in these components. The rest of the system’s structural components are then proportioned to behave elastically.

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Building codes largely adopt the R-factor design procedure . In this procedure, elastic seismic forces are reduced by factors Rd and Ro to obtain design forces. This procedure has matured into the “capacity design procedure”. The R-factor procedure can be illustrated by using a typical lateral load-deformation model. If we have a structure designed to behave elastically when subjected to an earthquake, the lateral loaddeformation relation would be linear and the structure would have a strength value corresponding to the elastic base shear ratio Ce caused by the earthquake, where Ce is a ratio of base shear force (V) caused by the earthquake to the weight of structure (W).

Ce =

V W

if it were elastic.

However, due to the uncertainties in determining the degree of severity of earthquakes and for obvious economical reasons, it is unrealistic to design a structure to respond elastically during a major earthquake. Thus, the code permits the reduction of the design base shear value to a lower value than the elastic base shear. The design base shear ratio defined by the code can be expressed as:

Cd =

Ce Rd Ro

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Cd =

Ce Rd Ro

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Two levels of reduction are inferred from this equation through the reduction factors

Rd and Ro :

Rd is a ductility-related force reduction factor that reflects the capability of a structure to dissipate energy through inelastic behavior.

Ro is an overstrength-related force reduction factor that reflects the dependable portion of reserve strength in the structure. Both of these factors form the essence of Canadian seismic design philosophy and have numerically assigned values in [Table 4.1.8.9] of NBCC-2010.

9.4.2 Seismic Lateral Force Resisting Systems The three most common systems used for seismic lateral force resistance are: Structural Shear Walls

Braced Frames

Moment-Resisting Frames

- no penetrations • resist lateral force as vertical cantilevers with rigid connection to the foundation

- triangular penetrations • resist lateral forces as a cantilevered vertical trusses (braced bays).

- rectangular penetrations • resist lateral forces through rigid connectivity between beams and columns (“rigid frames”).

One of the three lateral systems should be present in each orthogonal direction of the structure.

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9.5 Seismic Forces estimation based on NBCC-2010 9.5.1 Hazard and Design Spectra The procedure for estimating seismic-induced forces on buildings presented by the NBCC-2010 is based on “seismic hazard analysis” for different locations in Canada. As illustrated on the 2010 Seismic Hazard Map of Canada, each Canadian location has a certain degree of relative hazard. This is reflected in the quantification of seismic forces. When designing a structure to resist earthquakes, the design engineer should check the location of the structure and the corresponding degree of hazard.

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Hazard maps are further expressed in terms of spectral response accelerations Sa(t):

The above spectral accelerations are for periods of < 0.2, 0.5, 1.0 and 2.0 seconds>, respectively at a probability of 2% in 50 years, and for “firm ground” conditions (NBCC soil class C). Spectral acceleration contours are in “g”s (gravitational acceleration). Note that there are two “spectral response accelerations” in NBCC-2010: Sa(T) S(T)

“5% damped” spectral response acceleration (shown above), ( and as defined in NBCC-2010, Article 4.1.8.4, sentence 1)

“design” spectral response acceleration, ( as defined in NBCC-2010, Article 4.1.8.4, sentence 7, and determined from Sa(T) shown above). It is a common error for un-experienced engineers to fail to recognize this distinction. Therefore, in this class, a subscripted Sda(T) will be used in lieu of S(T).

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Using the Sa(T) information from seismic hazard maps which are expressed in terms of “damped” spectral response acceleration, the engineer can develop the “design” response spectrum for any locality in Canada. The design spectral acceleration (expressed as a fraction of gravitational acceleration), for a period T, is defined as Sda(T) and is given as: Fa Sa(0.2) for T 0.2 seconds, Fv Sa(0.5) or FaSa(0.2), whichever is smaller for T=0.5 sec, Fv Sa(1.0) for T=1.0 sec, Fv Sa (2.0) for T=2.0 sec, Fv ½Sa (2.0) for T 4.0 sec. where: T period (seconds) linear interpolation may be used for intermediate values, Sa(T) the damped spectral response acceleration from seismic hazard maps, Fa and Fv acceleration- and velocity-based site coefficients, respectively. Fa and Fv values are given in the NBCC-2010 and depend on the type of site condition (rock, or dense soil, or soft soil,.. etc). Sample design spectra for different localities are illustrated in the following figure. Sda(T)

= = = = =

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9.5.2 Base Shear Force Estimation For structures with geometrical and structural regularity, NBCC-2010 Article 4.1.8.11 permits using an Equivalent Static Force procedure for estimating the design base shear force. More rigorous dynamic analysis is required for structures with geometrical and/or structural irregularities, which are beyond the scope of the present chapter. For regular structures, Article 4.1.8.11 defines the minimum base shear force as:

V=

S (Ta ) M v I E W Rd Ro V=

S da (Ta ) M v I E W Rd Ro

where, V is the base shear force; Ta is the fundamental period of vibration of the structure; S(Ta) Sda(Ta) is the design spectral acceleration corresponding to the fundamental period Ta; Mv is a factor which accounts for higher mode effects; IE is a factor which accounts for the degree of importance of the structure; Rd and Ro are the ductility-, and overstrength- related force reduction factors, respectively; and W is the weight of the structure. The previous formula can be viewed as:

V=

Elastic Base Shear Coefficient W Force Reduction Factors

similar to Section 9.4.1. This approach can only be used for structures satisfying the conditions of Article 4.1.8.7.

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9.5.3 Distribution of Forces (along the height of the building). NBCC-2010 establishes the vertical distribution of the total base shear force at different floor levels based on the relationship:

Fx = ( V − Ft )

Wx hx n

Wi hi i =1

where:

Ft = 0.07TaV but does not to exceed 0.25V

and, Fx = the force at floor number xi V = the total base shear as defined in Section 9.5.2 Ft = force portion concentrated at the top of the building in addition to the above distribution, Wx = the weight of floor xi hx = the height of floor xi from the foundation level, and, n

Wi hi i =1

= summation of (floor weight) x (height) for all the floors in the building, and n =number of floors.

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9.6 Design of Ductile Moment-Resisting Frames based on S16-09 Clause 27 of S16-09 present both proportioning and detailing requirements which provide acceptable inelastic response of steel structures under seismic actions. The main objectives of Clause 27 code provisions are: a) avoid unstable sidesway mechanisms for structures exhibiting inelastic behavior. b) ensure ductile flexural behavior in yielding regions of the steel frame. In other words, S16-09 code provisions for seismic design provide the guidelines to correctly defining the locations of yielding regions (fuses, plastic hinges) as well as the criteria for detailing the steel frames to ensure a safe failure of these regions under the effect of a major earthquake level. This section highlights some of the principles presented by S16-09 for the design and detailing of Ductile Moment-Resisting Frames.

9.6.1 Strong-Column/Weak-Beam principle Clause 27.2.1.1 promotes a multi-storey side-sway mechanism dominated by hinging of beams rather than columns. The requirement of the formation of hinges (fuse locations) at beam ends and at column bases only is termed strong-column/weak-beam design. This is intended to avoid the formation of weak-storey (single-storey) mechanisms in which hinging occurs at the top and bottom ends of a single storey leading to overall instability.

weak-column / strong-beam

strong-column / weak-beam concept

To achieve strong-column/weak-beam design, S16-09 requires that the sum of columns flexural strengths at each joint exceed the sum of beam flexural strengths as specified by Clause 27.2.3.2.

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9.6.2 Beam-to-Column Connections S16-09 provisions require that a beam-to-column connection is capable of transferring the moment and shear forces developed in the beam to the column. Clause 27.2.5.1 requires that the connections should be capable of deforming in order that the frames can achieve specified drift levels. This is further discussed in Annex J where reference is made to pre-qualified connection configurations that have been tested for the ability to provide satisfactory performance. Some of these configurations are illustrated below: Reduced beam section connection

Bolted flange plate connection

Bolted bracket connections

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9.6.3 Detailing for Ductile Behavior The higher the level of ductility of the lateral load resisting system, the more it is expected to undergo significant inelastic behavior. Thus, S16-09 sets detailing provisions to ensure ductile behavior which include: Protected Zones: Clause 27.2.8 requires the designation of regions subject to inelastic deformations. Clause 27.1.9 sets the requirements for protected zones where structural and other attachments that can alter the desired behavior of these zones should be prohibited. Protected zones should be indicated on structural design documents and shop details.

Compact Sections (Class 2): To ensure reliable inelastic deformation, S16-09 requires width-to-thickness b/t ratios of compression elements to be limited such as to avoid local buckling (Class 1 or 2). Column splices: Since column splices are critical to the overall performance of moment-resisting frames, it is essential to ensure the reliability of the splice. In most cases a complete joint penetration grove weld is required for column splices. New and innovative concepts: The University of Toronto has developed a “yielding brace system” (YBS) shown in the attached photograph by Michael Gray. Details of this “scorpion YBS” system are given in CISC’s “No 41, Fall 2011” publication of ADVANTAGE STEEL.

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9.7 Seismic Example:

8000

9x3600=32400

8000

EQUIVALENT STATIC FORCE PROCEDURE

Level

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.

Storey Weight (kN)

9

1500

8

1800

7

1800

6

1800

5

1900

4

1870

3

1870

2

1870

1

2080

required: - vertical distribution of lateral seismic-induced forces due to “ground shaking” only according to ESFP “EQUIVALENT STATIC FORCE PROCEDURE) using the “equivalent lateral seismic force” Wx hx = ( − ) F V F “base shear” equation: x t n Article 4.1.8.11, sentence 6) Wi hi i =1

given: 9-storey office building has an SFRS (SEISMIC FORCE RESISTING SYSTEM) consisting of a steel ductile moment-resisting seismic frame as illustrated above, - located in Victoria, BC - rock soil condition equivalent to site soil “class B” solution steps: Note: all the information required to solve this problem is provided by NBCC2010 in Subsection 4.1.8 and its Articles 4.1.8.__ and its sentences 4.1.8.__ ) also, as stated in Articles 4.1.8.4 the 5% damped SRA data provided for the problem is based on Class C “reference ground conditions” for the locale in question. designer input required to solve the problem: “go to” …. in NBCC-2010 Ta fundamental period of lateral vibration of building in the direction of analysis. • Article 4.1.8.11, sentence 3) Spectral Response Accelerations SRA: • Article 4.1.8.4, sentence 1) ,or, • Sa(T) 5% damped SRA • seismic hazard maps, or • COMMENTARY J (Table J-2) • Sda (T) design SRA • Sda (Ta) design SRA at T=Ta

__________

• Article 4.1.8.4, sentence 7)

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step 1: estimate of “fundamental period” Ta : The response spectrum adopted by the Code is based on estimating a value of the fundamental period of vibration of the structure. NBCC-2010 provides a number of empirical formulas for different structural systems to estimate the fundamental period: Article 4.1.8.11 (sentence 3): 3) The fundamental lateral period, Ta in the direction under consideration, shall be determined as: a) for moment-resisting frames that resist 100% of the required lateral forces and where the frame is not enclosed by or adjoined by more rigid elements that would tend to prevent the frame from resisting lateral forces, and where hn is in metres: i) 0.085 (hn)3/4 for steel moment frames ii) 0.075 (hn)3/4 for concrete moment frames, or iii) 0.1 N for other moment frames, b) 0.025 hn for braced frames where hn is in metres, c) 0.05 (hn)3/4 for shear wall and other structures where hn is in metres, or d) other established methods of mechanics using a structural model that complies with the requirements of sentence 4.1.8.3.(8), except that: i) for moment-resisting frames, Ta shall not be taken greater than 1.5 times that determined in Clause (a), ii) for braced frames, Ta shall not be taken greater than 2.0 times that determined in Clause (b), iii) for shear wall structures, Ta shall not be taken greater than 2.0 times that determined in Clause (c), iv) for other structures, Ta shall not be taken greater than that determined in Clause (c), and v) for the purpose of calculating the deflections, the period without the upper limit specified in Subclauses (d)(i) to (d)(iv) may be used, except that for walls, coupled walls and wall-frame systems, Ta shall not exceed 4.0 sec, and for momentresisting frames, braced frames and other systems, Ta shall not exceed 2.0 sec. - the upper limits (above) are imposed on the periods of structures (Ta) because of concern that structural modeling does not include non-structural stiffening elements, thereby resulting in values of Ta which are too high results in calculated seismic design forces which will be too low. calculations: • Hn= 9 x 3.6 metres=32.4 metres from Article 4.1.8.11 (above), Ta= 0.085(32.4)3/4=1.15 seconds • the structure meets the criteria of Article 4.1.8.7, sentence 1), case (b) and, therefore qualifies for analysis by ESFP method (equivalent static force procedure).

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step 2: Spectral Response Accelerations (SRA): This is the crucial step in seismic analysis and is based on the hazard classification given by NBCC. The value of Sa(T) 5% damped spectral response acceleration will depend on the period of the structure and the location of the building. It can be obtained using the maps included earlier in this chapter or, in this case, from NBCC-2010 – COMMENTARY J (Table J-2). The “reference soil” is always “class C”. Some of the data from this table is: City Victoria Vancouver Calgary Edmonton Saskatoon Regina Winnipeg

Sa(0.2)

Sa(T): 5% damped SRA seismic data Sa(0.5) Sa(1.0) Sa(2.0) PGA

1.2 0.94 0.15

0.82 0.64 0.084

0.38 0.38 0.041

0.18 0.17 0.023

0.61 0.46 0.088

0.10 0.095

0.057 0.057

0.026 0.026

0.008 0.008

0.040 0.036

Since the period of our structure is t=1.15 second, we interpolate between the values of Sa(1.0) and Sa(2.0) for Victoria, BC and could get spectral response acceleration at t=1.15 sec, as Sa(1.15sec)= 0.35 (units are fractions of “g”), but there is a better way using the complete spectrum. 0.38 0.35 - also, Sa(0.2)= 1.2 > 0.12, ∴ according to Article 4.1.8.1, sentence 1) need to design for 0.18 0.19 Subsection 4.1.8 “Earthquake Load and Sa(1.15) Effects” ____________________________________ Sa(1.0) Sa(2.0) step 3: Sda (T) design Spectral Response Acceleration: - need values for Fa and Fv: and the Table below Article 4.1.8.4, sentence 7). = Sda (T) Fa • Sa(0.2) for T 0.2 sec. = Fv • Sa(0.5) or Fa• Sa(0.2), whichever is smaller for T=0.5 sec. = Fv • Sa(1.0) for T=1.0 sec. = Fv • Sa(2.0) for T=2.0 sec. = Fv • Sa(2.0)/2 for T 4.0 sec.

NEHRP

Fa

acceleration-based site soil coefficient; it is based on the “short-period” amplification factor of Sa(0.2) and it is required for “non-Class C” soils.

Fv

velocity-based site soil coefficient; it is based on the “long-period” amplification factor of Sa(1.0) and it is needed for “non-Class C” soils.

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- the site soil conditions are given as class B , the “reference soil is class C,

∴ need to use Tables 4.1.8.4 A, B & C to get the “site coefficients” Fa and Fv (for educational purposes, copies of these tables are reproduced below):

reference soil “C”

example site soil

example site soil

example: Sa(1.0)=0.38

Fv= 0.78

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To obtain seismic design parameters for site soil “class B”, use the above values of Fa and Fv from NBCC Tables B and C. • Table B has Fa at Sa(0.2) (“short period” vibrations), while • Table C has Fv at Sa(1.0) (“long period” vibrations), At “class C” sites, all values of Fa and Fv are unity (=1.0) because this is the “reference soil class” on which the “seismic hazard maps” are based but our example is “class B”. Calculations need to be done on the Sa( ) values provided by NBCC-2010 (Appendix C) to get S( ) Sda( ) for Victoria “B soils” as indicated below: city: Victoria, B.C. NBCC-2010 Appendix C Sa(T): 5% damped SRA (response spectrum) data from Seismic Hazard Maps (“class C” soils) - design response spectrum calculations for “class B” Victoria soils

S(T)

Sda(T)

Response Spectra (accelerations) T= 0.2 sec

Sa(0.2)=1.2

Fa • Sa(0.2) =(1.0)(1.2) = 1.2

T= 0.5 sec

Sa(0.5)=0.82

• lesser of: Fv • Sa(0.5) and

Fa• Sa(0.2), = (0.78)(0.82)

T= 1.0 sec Sa(1.0)=0.38

T= 2.0 sec Sa(2.0)=0.18 PGA=0.61

Fv • Sa(1.0)

Fv • Sa(2.0) =(0.78)(0.38) =(0.78)(0.18) = 0.296 = 0.140

or, (1.0)(1.2) = 0.640

To get the overall picture, the above Sa(T) and S(T) Sda(T) should be presented on graphs as shown below. This is the “design response spectrum” for the project and is used repeatedly for “static” as well as “dynamic” analyses of structures on the project. Class B soil Victoria, B.C.

T= 1.15 sec

Sda(1.15 sec)= 0.273

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!!

IX - 26

"

SRA summary for Victoria example: • site soil class “B” • Ta=1.15 sec, and Sa(1.15)=0.35 (interpolated, but not required if using graphs above), • Fa=1.0 at Sa(0.2), • Fv=0.78 at Sa(1.0), ∴ Sda(Ta) Fv • Sa(Ta) =0.273 as interpolated from graphs, or 0.78 x 0.35= 0.273

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step 4: Base Shear Design Force: (remember, in these notes: S(Ta) Article 4.1.8.11, sentence 2)

V=

S da (Ta ) M v I E Rd Ro

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Sda(Ta) )

W

- need more Tables to get IE and Mv but these are simple-to-read values as follows: • from Table 4.1.8.5, “ normal importance” IE =1.0 • from Table 4.1.8.11, “ higher-modes” for ductile MRFs (moment-resisting frames) with Sa(0.2)/Sa(2.0)=1.2/0.18= 6.7 Mv =1.0 Rd & Ro : • from Table 4.1.8.9, for ductile MRFs (moment-resisting frames) with IE Fa Sa(0.2)=(1.0)(1.0)(1.2)= 1.2 , read Rd =5.0, Ro =1.5 Note: a high ductility rating like Rd =5.0 will require “ ductile detailing and design” . Total Weight of Structure: = Wi=16490 kN Base Shear: V=0.273 x 1.0 x 1.0 x 16490 kN / (5.0 x 1.5) = 600 kN 3.6% W _________________________________________

step 5: check Minimum and Maximum Limits of Base Shear: • to safeguard against “ long period” repetitive sway vibrations where “ ductility demand” might not be “ uniform” along the height of the frame: Vmin: Article 4.1.8.11, sentence 2) case (b)

Vmin =

S ( 2 . 0) M v I E Rd Ro

W

Vmin(Class C)=(0.18)(1.0)(1.0)W/ (1.5x5.0) = 0.024W= 0.024 x 16490 kN= 396 kN Class B soil will be less yet (0.140) and ∴ Vmin does not govern. • for “ short period” vibrations the ESFP method overestimates the magnitude of base shear and, for SFRS systems with Rd 1.5, an “ experience-based” factor of is used to place an “ upper bound” on the value of V , see COMMENTARY J (page J-49). Vmax: Article 4.1.8.11, sentence 2) case (c)

Vmax

2 S (0.2) I E =3 W Rd Ro

for SFRS systems with Rd

1.5 and not on “ class F” soils.

class C soil Vmax =( )(1.2)(1.0)W/7.5=0.1067W=0.1067x16490 kN= 1759 kN class B soil Vmax same as “ class C” ∴ Vcalculated governs, use V= 600 kN. __________________________________________________

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step 6: Vertical distribution of horizontal forces: Article 4.1.8.11, sentence 6)

Fx = ( V − Ft )

Wx hx n

Wi hi i =1

where: Ft= 0.07 Ta V but not to exceed 0.25V Ft= 0.07(1.15)V= 0.081V= 48.3 kN (V− −Ft)= 600.0− 48.3=551.7 kN

Σ=

Level

hx[metres]

Wx[kN]

Wxhx[kN.m]

Wxhx/ Wihi

Fx [kN]

9 8 7 6 5 4 3 2 1

32.4 28.8 25.2 21.6 18 14.4 10.8 7.2 3.6

1500 1800 1800 1800 1900 1870 1870 1870 2080

48600 51840 45360 38880 34200 26928 20196 13464 7488

0.1693639 0.1806549 0.158073 0.1354912 0.119182 0.0938402 0.0703801 0.0469201 0.0260946

141.74 99.67 87.21 74.75 65.75 51.77 38.83 25.89 14.40

16490 [kN]

286956 [kN.m]

1.000000

600.0 [kN]

[end of ESFP example problem.]

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9.8 Seismic Analyses summary Structural seismic analyses is a world of its own. The structural design analysts may encounter any of the following procedures indicated below. National and international codes are trending away from ESFPs (equivalent static frame procedures). • Linear Static:

SEISMIC ANALYSIS / DESIGN TOOLS

o Building code formulae for calculating “base shears” similar to ESFP. (NBCC-2010 Article 4.1.8.11 , see example problem 9.7 class notes). • Linear Dynamic by Modal Response Spectrum Analysis (Article 4.1.8.12): o “response spectrum” analysis requires a response-spectrum curve consisting of digitized points of pseudo-spectral accelerations vs. time periods in a given direction for the structure (Commentary J, Note 32) o response spectrum seeks maximum response rather than a full time history analysis. It is based on modal superposition and eigenvectors or Ritz vectors. • Non-Linear Static: Pushover Analysis o performance-based analysis, o the structure is pushed to failure with an increasing load up to expected level of performance, o progression of plastic hinges are monitored until complete collapse mechanism is formed. • Time History Analyses (THAs) (NBCC-2010 Article 4.1.8.12): - used at fault locations and whenever historic data is available, o can be linear, or non-linear (geometric and/or material). o can be modal superposition (MTHA) or direct-integration . o can be transient or periodic. Although more sophisticated seismic analyses are a daunting task, they are becoming the requirement of codes. Software such as SAP2000 and related tutorials, webinars and training sessions are available to assist structural seismic engineers. __________________________________________[end of Chapter IX “ Seismic Loads” ] • prepared (2010) by: M.M. Safar, Ph.D. (McMaster), P.Eng. • edited (2012) by: A. Mir, M.Eng. (UBC), P.Eng. • reviewed by: M.M. Hrabok, Ph.D. (Alberta), P.Eng.