Evaluation of the Accidental Torsion Requirement in...

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BSSC SDC B E: Accidental Torsion Studies E-1

Appendix E

Evaluation of the Accidental Torsion Requirement in ASCE 7 by the FEMA P695 Methodology

E.1 Overview

The purpose of this study is to evaluate the significance of the accidental

torsion requirement in Section 12.8.4.2 of ASCE 7-10 for buildings in SDC

B. The accidental torsion provisions require application of a +/-5% offset of

the center of mass in each of two orthogonal directions to compute a

torsional moment, thereby increasing the design seismic base shear. The

primary goal of this study is to quantitatively examine the possible

elimination or revision of the accidental torsion requirement for SDC B

buildings designed according to the newly proposed stand-alone code

document. To this end, the study quantifies the effect of the accidental

torsion design requirement in terms of building collapse capacity and

collapse risk for a variety of SDC B buildings in order to determine the

consequences, or lack thereof, of removing or revising the accidental torsion

requirements.

Seismic ground motions may induce torsional response in buildings. Some of

this torsion is created by asymmetrical building geometry, hereafter referred

to as “inherent” torsion. In contrast, accidental torsion is unexpected and

may occur for a variety of reasons including: asymmetric distribution of

lateral ground motions across the plan of the building, asymmetric stiffness

contributions from the gravity system or nonstructural elements not

accounted for in design, uneven live-load distribution, or changes in the

center of rigidity due to nonlinear behavior. To account for all of these

potential sources of accidental torsion, ASCE 7 defines an accidental

torsional design moment to be considered in buildings with rigid diaphragms,

in addition to any inherent torsion that may exist. The additional accidental

torsional moment is equivalent to the torsion due to applying the seismic load

at a distance from the center of mass equal to 5% of the building dimension

perpendicular to the applied lateral load (Section 12.8.4.2 of ASCE 7-10).

These provisions have the effect of increasing the design base shear in the

frames and walls that resist lateral forces.

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E-2 E: Accidental Torsion Studies BSSC SDC B

E 1.1 Literature Review on Accidental Torsion

Significant research on accidental and inherent torsion in buildings has been

conducted in the past, producing varying results (Stathopoulos et al. 2009;

De Stefano and Pintucchi 2007). However, those that have specifically

addressed the issue of accidental torsion in design codes have all concluded

that it is not needed for most ‘regular’ buildings; ‘regular’ loosely refers to

buildings that are not particularly torsionally flexible or irregular, although

the exact definition varies from study to study (Stathopoulos et al. 2005,

Chang et. al 2009).

Some of the key findings of the past research are outlined here. De Stefano

and Pintucchi (2007) provide a more complete summary and assessment of

recent research on torsion in buildings. This section summarizes selected past

research of torsional seismic response of buildings, based on three main

categories of models: 1) linear models, 2) simplified single-story shear-

spring models and 3) lumped plasticity nonlinear frame models.

Results from linear models have shown that design accidental torsion is not a

significant factor for building performance under earthquake excitation for

many buildings (Chopra 1992. A strength of the linear models is that they

represent realistic building geometries, such as multi-story space frames,

very accurately. However, the linear models cannot accurately simulate post

yielding behavior and collapse and, more recently, detailed nonlinear models

have been used to evaluate torsional response.

Some of the first nonlinear models used for studying torsion in buildings

were models that represented the aggregated behavior of lateral force

resisting systems with bilinear shear springs in a single-story. These models

have the advantage of being able to simulate behavior beyond the linear

range. Anagnostopoulos et al. (2009) showed that the procedures for

calibrating such simplified models is crucial for obtaining accurate results.

They demonstrated that by calibrating single story shear spring models to

more high end lumped plasticity models using pushover analysis, they could

obtain results that qualitatively agreed with the results from the more

sophisticated nonlinear frame models with lumped plasticity elements. The

results of the single-story simplified models even with agreed lumped

plasticity models having more than one story (based on 3 and 5 story

models).

However, Anagnostopoulos et al. (2009) recommends that strong caution be

taken when calibrating simplified shear spring models. For example, some

researchers have scaled building strength of the simplified models based on

design loads, without modifying the stiffness. Increasing strength

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independently of stiffness in bilinear models leads to inaccurate measures of

initial stiffness, but also artificially increases the yield displacement. Since

much of the early work on torsion with nonlinear models used ductility

demand as a performance measure, artificially increasing the yield

displacement of a lateral force resisting system artificially increased its

performance; this phenomenon has been addressed quite specifically by Tso

and Smith (1999). Anagnostopoulos et al. (2009) showed that carefully

calibrated simplified shear-spring models predict greater ductility demand on

the flexible side of irregular buildings, which is in agreement with linear

models and lumped plasticity models. However, increasing strength

independently of stiffness in simplified models leads to the exact opposite

prediction; ductility demand is increased in the stiff side elements. Since

they used ductility demand as a main performance measure, the calibration of

the simplified models made the difference between being qualitatively right

or qualitatively wrong in their predictions.

More recent work using nonlinear models of frame structures includes

Stathopoulos and Anagnostopoulos (2009) and Chang et al. (2009).

Stathopoulos and Anagnostopoulos (2009) used one, three and five story RC

space frames with lumped plasticity models of beam and column elements to

assess the importance of design accidental torsion, concluding that it is

insignificant for the building heights and torsional rigidities studied and

ought to be re-examined. A similar study by Chang et al. (2009) examined

six and twenty-story steel space frames and reached the same conclusion;

design accidental torsion requirements are not significantly beneficial, for the

building types they studied. Both of these studies used ductility demand as

the main measure of performance.

This study will expand on past work by considering a wider variety of

torsional flexibility and irregularity in buildings, focusing on collapse

capacity as the primary performance metric. To our knowledge, no studies

of accidental torsion have used collapse as a performance measure, as we do

here, but instead have relied mostly on ductility demands to quantify the

impacts of accidental torsion (De Stefano and Pintucchi 2007). The building

designs considered here have base shear levels consistent with SDC B, in

contrast to most previous studies that considered SDC D ground motion

levels, and consider a broad range of torsional flexibilities.

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E.1.2 Methodology

In this study, we consider the effects of accidental torsion code provisions on

a set of archetype nonlinear building models, which include torsionally stiff

and torsionally flexible structures, and have ductility, mass, and strength

characteristics of SDC B buildings. Since a main purpose of seismic codes is

to reduce the likelihood of earthquake-induced collapse, accidental torsion

requirements are evaluated with regard to seismic collapse risk.

FEMA P695 proposes a methodology for systemically evaluating the seismic

design provisions of new seismic resisting lateral systems on the basis of

ensuring an acceptably low probability of collapse. The method uses

building collapse capacity as a metric for determining appropriate response

coefficients R, Cd, and Ω0 for newly proposed systems. The process for

implementing the FEMA P695 methodology is illustrated in Figure E-1.

Figure E-1 Flow chart schematic of FEMA P695 methodology

(FEMA 2009)

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In this study, the FEMA P695 method has been adapted to evaluate a

particular code provision, namely the 5% offset requirement to account for

accidental torsion, rather than a specific Seismic Force Resisting System

(SFRS), but the main concepts have not changed. Rather than focus on a

specific system, the method has been used to evaluate the collapse

performance of a set of typical SDC B buildings designed with and without

the accidental torsion requirement. To this end, an archetype design space is

developed, analytical models created and analyzed, and their collapse

performance is evaluated. The difference in collapse risk with and without

accidental torsion provides quantitative information as to the importance of

including accidental torsion requirements in SDC B. Each of these steps is

documented in detail in the following sections.

E.2 Archetype Design Space

The objective of this study is to quantify the effect of the accidental torsion

requirement on the design and safety of buildings in SDC B. Therefore, it is

important to identify a range of archetype designs that encompass as many

SDC B buildings as possible, with special emphasis on those buildings that

may be most affected by accidental torsion requirements. This section

discusses building characteristics that may affect the influence of accidental

torsion requirements in design and how these characteristics were considered

in developing a representative set of buildings. Table E-3 and

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Table E-4, at the end of this section, summarize the suite of archetype

designs that are analyzed. Every archetype building is designed in two

versions: one with and one without the accidental torsion design requirement

considered, to provide a direct assessment of the impacts of accidental

torsion design requirements on building collapse performance. The archetype

design models were created by calibrating their linear and nonlinear

properties to a set subset of baseline ‘high end’ OMF frame models.

E.2.1 Seismic Force Resisting System (SFRS)

Building systems most commonly used in SDC B are less ductile than those

used in higher seismic design categories. In fact, most have values of R, the

response modification coefficient, of around 3. Due to the infeasibility of

analyzing every available SFRS for SDC B, the models in this study are

based on the design and behavior of reinforced concrete Ordinary Moment

Frame (OMF) models. The choice of OMFs to represent SDC B buildings

more generally is justified by this study’s focus on measuring the effect of

designing for accidental torsion on collapse capacity and collapse risk, not

comparing specific systems. Reinforced concrete OMFs are used because

they are non-ductile, their nonlinear behavior is fairly well documented and

modelable, and they are commonly used in SDC B. In addition, the most

important properties pertaining to collapse capacity such as ductility,

overstrength, and deformation capacity are fairly similar to many other

systems used in SDC B.

E.2.2 Building Height

Three different building heights are used in this study in order to capture the

effects of designing for accidental torsion: 1, 4, and 10 stories. The height of

10 stories (132 ft.) was chosen as the tallest archetype structure because it is

tall enough to adequately capture the effects of higher modes in tall

buildings. Past studies by Chang et al. (2009) and Stathopoulos and

Anagnostopoulos (2009) have suggested that accidental torsion requirements

are less beneficial for taller buildings (5, 6, and 20 stories) than single story

buildings.

E.2.3 Building Weight

Since gravity loads can play a major role in the design of SDC B buildings, a

range of building weights are considered. The ‘low’ and ‘high’ gravity

scenarios in this study are 100 psf and 200 psf of un-factored dead weight,

respectively, for all stories except the roof level. ‘Low’ and ‘high’ roof

weights are 80 psf and 160 psf, respectively, and are used for the single-story

buildings. These values are intended to represent a reasonable range of

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BSSC SDC B E: Accidental Torsion Studies E-7 S2

S1=2S2

L2=100'

Frame 3

Frame 2 Frame 4

weights of buildings, but are not linked to any particular floor system or

occupancy. Live load was taken to be 20 psf at the roof level and 50 psf for

all other stories, and live load reductions were made according to section 4.7

of ASCE 7-10 Past research has shown that gravity load levels can

significantly affect system ductility, overstrength, and collapse performance

(FEMA 2009).

Only the high gravity load level was used for the 10-story archetype designs

because the 1-story and 4-story archetypes showed that high gravity

buildings performed worse overall and had more significant improvements

from design accidental torsion than their low gravity counterparts .

E.2.4 Building Plan Layout

Most of the archetype building layouts considered in this study are

symmetric (rectangular layouts). Past research (Llera and Chopra 1995,

Stathopoulos and Anognostopoulos 2005), have shown that accidental

torsion requirements have a larger effect on the performance of symmetric

buildings than asymmetric buildings, because the relative increase in

torsional design forces due to accidental torsion increases as inherent torsion

decreases. In addition, we consider buildings with different torsional

rigidities because the torsional period or frequency affects response to

earthquake excitation.

The rectangular building plans follow the schematic in Figure E-2Error! Not

a valid bookmark self-reference., with overall building dimensions 200 ft.

x 100 ft. and relative frame spacing of S/L=S1/L1=S2/L2= 1.0, 0.75, 0.5, and

0.25. This configuration is used for all of the rectangular buildings in this

study. (In the context of this study, the term “frame” refers to any frame or

wall line that is part of the lateral force resisting system). The extent to which

designing for accidental torsion increases the design base shear in frame lines

depends on the relative torsional stiffness of a structure and its frame-line

spacing. This effect is illustrated using the building plan that is illustrated in

Figure E-2. The building has plan dimensions L1 and L2 and frames are

spaced at distances S1 and S2 apart. All frames are considered to have equal

stiffness k. Taking a normalized design base shear of 1 in each frame and

then computing the additional shear due to accidental torsion produces the

results shown in Table E-1. In general, as relative frame spacing decreases,

torsional rigidity decreases and the contribution of accidental torsion to the

design base shear in frames increases.

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Figure E-2 Plan view of a symmetric archetype structure with a

rectangular frame layout

Table ETable ETable ETable E----1111 Increase in Increase in Increase in Increase in BBBBase ase ase ase SSSShear hear hear hear DDDDue to ue to ue to ue to the the the the 5% 5% 5% 5% OOOOffset ffset ffset ffset AAAAccidental ccidental ccidental ccidental

TTTTorsion orsion orsion orsion RRRRequirementequirementequirementequirement for Building for Building for Building for Building Layout Layout Layout Layout shown in shown in shown in shown in

Figure EFigure EFigure EFigure E----2222

S/L

Frames 1&3 Frames 2&4

Design Base

Shear

(Normalized)

Total Design Base

Shear, Accounting for

Accidental Torsion

Design Base

Shear

(Normalized)

Total Design Base

Shear, Accounting

for

Accidental Torsion

1

(Perimeter) 1 1.02 1 1.08

0.75 1 1.03 1 1.11

0.5 1 1.04 1 1.16

0.25 1 1.08 1 1.32

In addition to the rectangular frame layout that was used most for most of the

archetypes analyzed, a subset of archetypes with an ‘I-shaped’ frame layout

was also analyzed. I-shaped or similar frame layouts are common in parking

garages and other structures.

L2=100'

S1

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Figure E-3 ‘I-shaped’ frame layout

E.2.5 Building Plans with Inherent Torsion

A few selected archetype buildings were analyzed with asymmetric building

plan layouts, as depicted in Figure E-4. Two different inherent torsion plan

layouts were used: one with S/L=S1/L1=S2/L2=0.5 and one with

S/L=S1/L1=S2/L2=0.25. For each of these layouts, two of the frames are

located at the building’s edge and the other two are inset according to the

prescribed relative frame spacing. Eccentricities in each direction are labeled

as e1 and e2. Both of the archetype building geometries used to represent

buildings with inherent torsion are classified as having horizontal irregularity

type 1b (extreme torsional irregularity) according to ASCE 7-10.

Figure E-4 ‘Inherent torsion’ frame layout

E.2.6 Natural Accidental Torsion

We use the term “natural accidental torsion” to describe the effective offset

between center of mass and center of stiffness, accounting for the many

sources of ‘accidental torsion’ that may exist. Levels of natural accidental

torsion were systematically introduced in the model, but not the design, by

offsetting the center of mass (CM) of the models from the design CM along

L1=2L

2=200'

L2=100'

S2

S1=2S

2

CM X

e1

e2 CR X

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the diagonal of the building. Center of mass offset distances of 0%, 5%, and

10% of the total diagonal length of the building were used.

E.2.8 Design Assumptions and Methodology for OMF Models

A subset of the archetype buildings was designed as reinforced concrete

OMF’s according to ASCE 7-10 and ACI 318-10 and are listed in Table E-2.

Each archetype building was designed for dead, live, and seismic loads using

all applicable load combinations; additional loading from snow and wind

were not considered. The design short period and one-second spectral

accelerations were taken as the maximum allowable values for SDC B:

SDS=0.33(g) and SD1=0.133(g).

The buildings were designed as space frames with 2-way slabs, having spans

of 30 ft., and story heights of 15 ft. and 13 ft. in the first story and all other

stories, respectively. For design, they were modeled as 2D portal frames with

SAP2000, using the Equivalent Lateral Force Procedure (ELFP) to determine

design loads, story forces and drifts. The design of all members was force

controlled, with the exception of the ten story archetypes whose lowest six

stories were governed by the stability (P-∆∆∆∆) requirements of Section 12.8.7 in

ASCE 7. Columns of the one-story buildings were modeled pinned at the

base, whereas all other designs used a fixed foundation assumption for

design, to be consistent with common design practice.

Each OMF design depended on the number of stories and gravity loads and

had two versions, which are summarized in Table E-2. The first version was

designed as a space frame with 30 ft. bays and 30 ft. of tributary width and

an equivalent tributary seismic mass. Space frame OMFs were selected

because they are common and have nonlinear behavior that we believe is

representative of many SDC B type buildings. This design ignores accidental

torsion effects, i.e. it is designed only for the base shear calculated according

to the equivalent lateral force method. These are later referred to as the ‘low

base shear’ models because they have the lowest design base shear of all

designs for their particular height and gravity load levels. In Table E-2, the

low base shear designs are the odd numbered designs. The second version

was designed with the same geometry and loads, except with larger design

base shear due to the consideration of 5% accidental torsion (later referred to

as the ‘high base shear’ models). For symmetric archetypes, the increase was

32%, which was due to the base shear increase from accidental torsion when

relative wall spacing (S/L) is 0.25 in a building with the geometry shown in

Figure E-2. Frames with extreme values of design base shear were selected

for design (even numbered designs in Table E-2), because simplified models

are later calibrated by interpolation of properties between ‘high end’ OMF

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models. In addition, for select archetypes with inherent torsion, additional

‘high end’ OMF frames were designed and modeled considering more

extreme changes in design base shear.

The high-end OMF models are designed as space frames with 30 ft. of

tributary width, but the 3D frame layouts have just two frame lines in each

direction and plan dimensions of 200 ft. x 100 ft., as shown in Figure E-2.

Note that these simplified models have only two frame lines in each

orthogonal direction to more easily capture a wide range of torsional

flexibilities, creating a discrepancy with the original OMF space frame

design. The discrepancy between the two building plans was reconciled by

adjusting the mass and weight of the 3D models to reflect the correct

building mass and weight tributary to just two of the OMF frames.

Since gravity loads contribute significantly to the frame element design

moments and forces, much care was taken to design the two versions of each

frame consistently. For each OMF, the lower base shear version was

designed first. Columns were designed to be as small as possible while

keeping the longitudinal reinforcement ratio below about 4.5%. Beams were

designed as T-beams, but with smaller longitudinal reinforcement ratios

(2.5%-3%) than columns. The beam longitudinal reinforcement ratios were

often governed by maximum reinforcement requirements (which limit

reinforcement and promote steel yielding before concrete crushing).

Transverse shear reinforcement was designed with bar sizes ranging from #3

to #5, and bar size was kept consistent for all columns and for all beams

throughout each building; in every case, rebar size was determined such that

the maximum allowable spacing could be used for all or the elements of the

frame, reflecting common engineering practice.

After designing the frame with the lower base shear, the high base shear

version of the same frame was designed. Starting with the first design,

element sizes and reinforcement were increased to accommodate the larger

loads. We aimed to keep reinforcement ratios as similar as possible by

increasing the reinforcement and element sizes concurrently.

Table ETable ETable ETable E----2222 Matrix of Matrix of Matrix of Matrix of OMFOMFOMFOMF DDDDesignsesignsesignsesigns ((((BBBBaseline aseline aseline aseline MMMModels)odels)odels)odels)

Design

#

Building

Height

(stories)

Lateral

System

Gravity

(Story

Weight)

Relative

Frame

Spacing

(S/L)

Inherent

torsion

Design

Accidental

Torsion

1 1

Concrete

OMF 80 psf

1* None

None

2 0.25 5%

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Design

#

Building

Height

(stories)

Lateral

System

Gravity

(Story

Weight)

Relative

Frame

Spacing

(S/L)

Inherent

torsion

Design

Accidental

Torsion

3 160 psf

1* None

4 0.25 5%

5

4

100 psf 1* None

6 0.25 5%

7 200 psf

1* None

8 0.25 5%

9 10 200 psf

1* None

10 0.25 5%

*Frame spacing does not matter if the building is symmetric and accidental torsion is

not considered

E.2.8 Design Assumptions and Methodology for Simplified Frame Models

Simplified models have been constructed such that the design lateral

earthquake force in each frame, without considering accidental torsion, is

exactly the same as the baseline case for the corresponding ‘high end’ OMF

model, so that their nonlinear properties can be matched directly. For

simplified archetypes designed for accidental torsion, the earthquake forces

are increased and frame properties are obtained by interpolation between the

low and high base shear versions of the ‘high end’ OMF frames. This

process is described in more detail in section E.4.

E.2.9 Archetype Design Space Tables

Table E-3 summarizes key properties of the archetype design space that has

been used for this study.

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Table E-4 lists all the buildings and design properties considered in the study,

including a total of 196 archetypical models.

Table ETable ETable ETable E----3333 Summary of Summary of Summary of Summary of Archetype Design SpaceArchetype Design SpaceArchetype Design SpaceArchetype Design Space

Design #

Building

Height

(stories)

Lateral

System

Gravity

(Story

Weight)

Relative

Frame

Spacing

(S/L)

Configuration Inherent

Torsion

Design

Accidental

Torsion

Natural

Accidental

Torsion

196 Total

Archetypes

1

4

10

Concrete

OMF

Low

High

1

0.75

0.5

0.25

*0.45

*0.4

*0.35

*0.3

Rectangular

Frame

Layout

*I-Shaped

Frame

Layout

None

(Torsionally

Symmetric)

*25%

0%

5%

0%

5%

10%

*Properties only represented by selected subgroups of the archetype design space

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Table ETable ETable ETable E----4444 Full Archetype Design SpaceFull Archetype Design SpaceFull Archetype Design SpaceFull Archetype Design Space

Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

1

80 psf

1

Rectangular

Frame Layout

None

(Symmetric)

No

0

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

*0

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

5%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

10%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

160 psf

1 Rectangular

Frame Layout

None

(Symmetric)

No

0

0.75

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Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

1

160 psf

0.5

Rectangular

Frame Layout

None

(Symmetric)

No

0

0.25

1

Yes 0.75

0.5

0.25

1

No

*0

0.75

0.5

0.45

0.4

0.35

0.3

0.25

1

Yes

0.75

0.5

0.45

0.4

0.35

0.3

0.25

1

No

5%

0.75

0.5

0.45

0.4

0.35

0.3

0.25

1

Yes

0.75

0.5

0.45

0.4

0.35

0.3

41.86% Draft


Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

1

160 psf

0.25

Rectangular

Frame Layout

None

(Symmetric)

Yes 5%

1

No

10%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

160 psf

0.5

I- shape

None

(symmetric)

No

*0

0.45

0.4

0.35

0.3

0.25

0.5

Yes

0.45

0.4

0.35

0.3

0.25

0.5

No

5%

0.45

0.4

0.35

0.3

0.25

0.5

Yes

0.45

0.4

0.35

0.3

0.25

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Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

1 160 psf

0.5

Rectangular

High Inherent

Torsion

(Extremely

Asymmetric)

No

*0 0.25

0.5 Yes

0.25

0.5 No

+5% 0.25

0.5 Yes

0.25

0.5 No

+10% 0.25

0.5 Yes

0.25

0.5 No

-5% 0.25

0.5 Yes

0.25

0.5 No

-10% 0.25

0.5 Yes

0.25

4

100 psf

1

Rectangular

None

(symmetric)

No

*0

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

5%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

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Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

4

100 psf

1

Rectangular

None

(symmetric)

No

10%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

4 200 psf

1

Rectangular None

(symmetric)

No

*0

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

5%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

10%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

10

200 psf

1

Rectangular

None

(symmetric)

No

*0

0.75

0.5

0.25

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Building

Height

(stories)

Gravity

(Story

Weight)

Relative

Frame

spacing

(S/L)

LRFS

Configuration

Inherent

Torsion

Design for

Accidental

Torsion

Natural

Eccentricity

(Offset of

CM)

10

200 psf

1

Rectangular

None

(symmetric)

Yes

*0 0.75

0.5

0.25

1

No

5%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

1

No

10%

0.75

0.5

0.25

1

Yes 0.75

0.5

0.25

* The natural eccentricity is zero, but small amounts of torsion are introduced due to

the nature of the simplified frame models (this occurs for any kind of frame in 3

dimensions)2.

E3 Analysis Procedure

E3.1 Ground Motions

This study uses a set of 22 pairs of far-field strong ground motions selected

by the FEMA P695 project. These motions are recorded from large

magnitude events at moderate fault rupture distances. Although there are no

ground motions in the far-field set from SDC B-like environments, the

FEMA P695 strong ground motion set is used without modification because

it: (1) provides a consistent ground motion record set through which to

examine relative changes in collapse capacity due to accidental torsion

requirements, and (2) contains broadband frequency content, which is

41.86% Draft


important for obtaining unbiased results for multiple buildings with varying

lateral and torsional periods.

In incremental dynamic analysis of the two-dimensional models, each

component of each of the 22 ground motions was applied, leading to a total

of 44 records scaled until collapse occurs. Ground motions were applied bi-

directionally and simultaneously to the three-dimensional models. Each

analysis was repeated twice for each of the 22 pairs of ground motions: once

with the north-south (NS) component acting along the x-axis of the building

and the east-west (EW) component acting along the y-axis, then again with

the components switched so that the NS and EW components acted along the

y-axis and x-axis, respectively. All of the results from the 44 cases were

used for computing collapse statistics, per FEMA P695.

E3.2 Incremental Dynamic Analysis

Ground motions are scaled to increasing intensities until collapse occurs for

incremental dynamic analysis. In this study, ground motion scaling is based

on the geometric mean1 of the spectral acceleration of the two components at

a specific building period, i.e. Sa(T1). The fundamental period of the model,

obtained from eigenvalue analysis, was used for scaling ground motions for

all two-dimensional models. Periods of the three-dimensional designs and

models vary slightly (10% or less) depending on how much the design base

shear is increased to account for accidental torsion; however, it is desirable to

use the same period for scaling ground motions such that results can be

directly compared to one another. Therefore, one representative period has

been selected to scale ground motions for each combination of height and

gravity load level that is used.

Once incremental dynamic analysis is performed, two statistical measures of

collapse performance are used: the Adjusted Collapse Margin Ratio (ACMR)

and probability of collapse given the maximum considered earthquake

(MCE) ground motion intensity level, denoted P(Collapse|MCE).

The maximum considered earthquake ground motion intensity (MCE) in

ASCE 7-10 is based on a target risk of 1% probability of collapse in 50

years. At many locations, the risk-targeted MCE is similar to a ground

motion intensity whose likelihood of occurrence corresponds to a 2%

probability of occurring in a 50 year time period (approximately a 2500 year

return period) at a site.

1 This scaling procedure is slightly different than the FEMA P695 method, which

scales a set of pre-normalized records together, but the end result of either method, in

terms of the assessed margin against earthquake-induced collapse, is expected to be

indistinguishable from the other (FEMA P695).

50% Draft


To compute the ACMR of a building, the Collapse Margin Ratio (CMR)

must be computed first, based on the ratio of the median collapse capacity, or

spectral acceleration causing collapse in incremental dynamic analysis, to the

MCE spectral acceleration at the site of interest as in:

CMR = Sacollapse,median(T1)/SaMCE (T1) (E.1)

In addition, Baker and Cornell (2006) have shown that rare ground motions

tend to have a different spectral shape than the ASCE code-defined design

spectrum; in fact, the spectra tend to have peaks at the period of interest.

Therefore, analysis using broadband sets of ground motions, such as the

FEMA P695 far-field set, which do not have the expected peaks and valleys

in the response spectra, yield conservative estimates of median ground

motion intensity at which collapse occurs. To account for the frequency

content of the ground motion set, the FEMA P695 methodology uses a

spectral shape factor (SSF) to adjust the CMR. The spectral shape factor is

based on the site hazard of interest and a building’s period and ductility and

ranges between 1.1 and 1.2 for the SDC B structures in this study. These

factors have been calibrated to adjust the CMR to the value that would be

obtained if ground motions with the appropriate spectral shape were selected

specifically for the building, rather than using a general set. The equation for

ACMR of 3-dimensional buildings is:

ACMR = 1.2 x SSF x CMR (E.2)

Tables of SSF values and a more detailed description of how to compute SSF

and ACMR can be found in Chapter 7 of FEMA P695. The 1.2 factor adjusts

three-dimensional model results to a two-dimensional equivalent collapse

capacity, as described in FEMA P695.

Since ACMR corresponds to a median collapse value that is scaled by MCE,

a collapse cumulative distribution can be constructed if the dispersion in the

spectral intensity at which collapse occurs is known. Chapter 7 of the FEMA

P695 report gives a detailed explanation of important factors such as

uncertainty in design and modeling properties that contribute to total collapse

dispersion, as well as how to combine them to obtain total collapse

dispersion (βTOT), quantified by the logarithmic standard deviation. Several

tables of pre-computed dispersion values for different combinations of model

quality, quality of design requirements, and quality of system test data are

also presented in FEMA P695, Chapter 7. Values of βTOT can vary from

0.275 to 0.95, but are mostly between 0.45 and 0.7. For this study, a typical

value of the total dispersion βTOT was assumed to be 0.65, based on the tables

in chapter 7 of FEMA P695. It should be noted, however, that factors such

41.86% Draft


as model quality and quality of design requirements are subjective, and

therefore, our selection of βTOT =0.65 was somewhat subjective as well.

The probability of collapse given MCE is computed from cumulative

distribution function that is defined by the adjusted collapse margin ratio

(ACMR) and the total logarithmic dispersion (βTOT) as follows:

P(Collaspe|MCE)=LognormalCDF(1,ACMR, βTOT) (E.3)

E.4 Nonlinear Modeling

E.4.1 Overview of Modeling Approach

The majority of the analysis for this study of accidental torsion relies on

simplified models, which have been calibrated to the fully designed OMF

buildings and models. The following steps outline the general method used

for building simplified models: 1) Build and analyze ‘high end’ OMF 2D

models of archetypes in Table E-2, 2) Calibrate simplified models to match

the 2D OMF behavior, and 3) Build simplified 3D models for all archetypes

in Table E-4 using the 2D frames. Each of these steps is discussed in more

detail in the following sections.

E.4.2 ‘High End’ OMF Models

Each of the fully designed OMFs (listed in Table E-2) was modeled as a

moment frame in OpenSEES (Open Source Earthquake Engineering

Software). Columns and beams were modeled using a lumped plasticity

approach, with plastic hinge properties of beams and columns computed

according to empirical relationships developed by Haselton et al. (2008).

These relationships are based on the design properties of the beams and

columns (i.e. concrete compression strength, element dimensions, axial load

ratio, and reinforcement detailing) and are therefore capable of representing

the influence of changes in design on the element modeling. Plastic hinges

were modeled using the ‘Ibarra Material’ in OpenSEES developed by Ibarra

et. al (2005). The Ibarra hinge materials have tri-linear monotonic backbones

and incorporate cyclic and in-cycle deterioration, which are important for

modeling collapse.

Shear failure is not modeled directly in the ‘high end’ models. However,

shear failure has been accounted for by means of a non-simulated collapse

mechanism. The non-simulated collapse mechanism is triggered by post-

processing of dynamic analysis results and depends on the column deflection.

Physically, the non-simulated collapse mode represents the loss of vertical

load carrying capacity in at least one column due to shear failure. Non-

simulated collapse modes are described in more detail in section E.4.5.

50% Draft


Beam/Column

P-∆ Truss

Beam/Column Plastic Hinge

Nonlinear Joint

In addition to plastic hinges in the beams and columns, nonlinear joint

behavior was modeled using 2D shear panels with an Ibarra pinching

material. Nonlinear joint properties were obtained from Lowes and

Altoontash (Altoontash 2004; Lowes et al. 2004). The primary factors

affecting joint strength and/or ductility are confinement, joint area, and

column axial load ratio. Many of the outer joints of the ‘high end’ models

failed during analysis, but failure of the interior joints was prevented, which

is what we expect in interior space frames, due to the high level of

confinement of interior joints.

Distributed gravity loads were applied to the beams, and all remaining dead

loads were applied to P-∆ columns, connected to the frame by rigid truss

elements. Building mass was lumped at the joints and foundation

connectivity was modeled as pinned in the 1-story models and fixed for the

others. (Since 4-story fixed and grade-beam foundation models resulted

nearly identical computed CMRs, these foundation fixities were judged to be

reasonable.)

The ‘high end’ OMFs were analyzed using Incremental Dynamic Analysis

(IDA) and static pushover analysis, and the results of each were used to

calibrate simplified models.

Figure E-5 Schematic of a four-story OMF model

41.86% Draft


E.4.3 Simplified Model Calibration Procedure

For each ‘high end’ OMF model, a simplified 2D model was made that

matched its properties as exactly as possible. The simplified models are

single bay x-braced frames with nonlinear braces, as shown in Figure E-6.

The braces are truss elements with hysteretic material properties defined by

the nonlinear Ibarra material. Like the nonlinear hinge materials in the ‘high

end’ models, the brace materials are characterized by a tri-linear monotonic

backbone and different modes of cyclic and in-cycle deterioration properties.

The properties of the tri-linear backbones were calibrated to the ‘high end’

models, as described in the following paragraphs. The columns of the

simplified models are rigid beam/columns; multi-story simplified models

have elastoplastic hinges in columns between the stories to allow for story-

story interaction to occur as it would in a moment frame structure. P-∆ loads

for the 2D simplified models were applied directly to the columns.

Figure E-6 Schematic of a four-story simplified model

The first step for calibrating the simplified 2D models was to match the static

pushover properties of the corresponding ‘high end’ 2D models, with P-∆

effects included in the analysis. This calibration was achieved by modifying

the brace properties, specifically initial stiffness, strength, hardening

stiffness, capping displacement and negative post-capping slope, until the

pushover analysis results of each story of the simplified and ‘high end’ OMF

models matched as nearly as possible. After matching the story by story

pushover analysis results, the pushover results of the building as a whole, as

well as modal periods, were checked to ensure that the overall static behavior

of the simplified models matched the behavior of the ‘high end’ OMF

models as closely as possible. Figure E-7 illustrates the pushover calibration

Rigid Beam/Column Element

Rigid Truss Element

Nonlinear Truss Element

Elastoplastic Hinge

50% Draft


comparison for the 2D, 4-story, high gravity archetype designed without

accidental torsion.

All of the simplified model properties except for cyclic deterioration

parameters were calibrated using static pushover. Lastly, the cyclic

deterioration properties of the simplified models were adjusted until the IDA

results matched the IDA results of the corresponding ‘high end’ model. Table

E-5 illustrates the IDA comparison between the two models.

One difficulty with calibrating simplified braced frame models to represent

the ‘high end’ OMF models was the inherent lack of story-to-story

interaction in the simplified models. If all column and beam elements are

modeled as truss elements, each story of the simplified braced frame

assemblies behaves independently of the stories above and below. Two

major problems arise from this behavior: higher mode periods are much

different for the simplified models than the high end models, and damage

concentrates in just one story during pushover and dynamic analysis, rather

than distributing to multiple stories. This problem has been remedied by

making the columns flexurally rigid and adding plastic hinges between

stories to simulate the story-to-story interaction that occurs in the OMF

frames. Plastic hinge properties in the simplified models are based on beam

and column properties in the corresponding OMF frames. As a result, higher

modes of the simplified models matched those of the ‘high end’ models and

earthquake damage was distributed to multiple stories in a similar manner as

well. Table E-5 shows a comparison of the first 3 modal periods for the

‘high end’ and simplified versions of the 4-story high gravity OMF

archetype.

41.86% Draft


Figure E-7 Static pushover results for the 2D, 4-story high gravity

model designed without accidental torsion and

analyzed using a triangular loading pattern with P-∆

effects considered

Table ETable ETable ETable E----5555 IDA IDA IDA IDA RRRResults for esults for esults for esults for the 2D, the 2D, the 2D, the 2D, 4444----storystorystorystory High Gravity High Gravity High Gravity High Gravity

Archetypical Model Designed without Accidental Archetypical Model Designed without Accidental Archetypical Model Designed without Accidental Archetypical Model Designed without Accidental

TorsionTorsionTorsionTorsion

Measure OMF Simplified Difference

Period (sec) 2.36 2.36 0.1%

Median Sacollapse (g) 0.189 0.191 1.3%

βtotal 0.65 0.65 NA

CMR 2.2 2.3 1.3%

ACMR 2.7 2.7 1.3%

P(Collapse|MCE) 0.064 0.062 -3.7%

Table ETable ETable ETable E----6666 Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4----story High Gravity OMF story High Gravity OMF story High Gravity OMF story High Gravity OMF

Archetype without Accidental Torsion ConsideredArchetype without Accidental Torsion ConsideredArchetype without Accidental Torsion ConsideredArchetype without Accidental Torsion Considered

Mode Period (s)

Difference 'High end' Simplified

1 2.36 2.36 0.0%

0 5 10 15 20 25 300

50

100

150

200

250

Roof Displacement (in)

To

tal

Ba

se

Sh

ea

r (k

ips

)

OMF

Simplif ied

50% Draft


Mode Period (s)

Difference 'High end' Simplified

2 0.86 0.84 -2.2%

3 0.50 0.52 3.4%

4 0.32 0.36 11.6%

Once the 2D behavior of the simplified models was calibrated to the ‘high

end’ 2D OMF models, 3D simplified models were created. These models

reflect the design plan dimensions of 200ft. x 100ft. There are two frame

lines in each orthogonal direction of the simplified models and one leaning

column in the center of each quadrant of the building to transmit P-∆ forces

to the rigid diaphragm. The P-∆ columns in the 3D models are not a part of

the frames like they are in the simplified 2D models; the reason for this

difference is because real buildings typically have gravity carrying elements

that are distributied fairly evenly throughout the building, not just in the

lateral system. Therefore, P-∆ columns have been placed at the center of

each quadrant in order for P-∆ forces to have an appropriate lever arm for

impacting torsional response. The thick black lines in Figure E-8 represent

the frame lines of a sample 3D model (each frame is modeled as shown in

Figure E-6, except that they no longer carry P-∆ loads) and the squares

indicate P-∆ columns.

Figure E-8 Plan layout of a 3D simplified model

Determination of the 3D brace frame properties was based on the design base

shear of the structure. For cases where the frames in the 3D models had

exactly the same design base shear as the frames in the 2D model, the

modeled frames were identical. For cases where the design base shear due to

accidental torsion was different, because of the building of interest did not

fall in the subset of archetypes fully designed as 2D frames OMFs, the

properties of the braces (and plastic hinge elements between stories for multi-

L1=2L

2=200'

L2=100' X CM, CR

X CM +10%

41.86% Draft


story buildings) were computed using linear interpolation between the high

and low base shear versions of the 2D frames. Model strength, stiffness and

cyclic deterioration parameters were interpolated based on the design base

shear of the frames. Such interpolations were only performed between

frames that had the same gravity load and number of stories.

Using interpolation to compute the frame properties meant that several

archetype buildings could be modeled in 3-D using only two fully designed

baseline archetypes for each combination of height and gravity load level. It

should be noted that the capping displacement of the calibrated 2D simplified

models was always determined such that no interpolation would be needed to

compute capping displacement for intermediate models. In other words, the

capping displacement of the high base shear version of a given archetype was

kept the same as the capping displacement of the low base shear version.

The reason that capping displacement was kept constant for each archetype is

because we believe that system ductility should be independent of design

base shear. Therefore, linking capping displacement to design base shear

would introduce error into the experiment by calibrating intermediate models

to design idiosyncrasies, rather than meaningful system properties.

Additionally, the capping displacemt for the high and low base shear

versions of each ‘high end’ OMF frame in this study were extremely similar

(consistantly less than10% different), which confirmed our decision to keep

it constant during calibration. An example of the interpolation of simplified

frame properties is shown below in Figure E-9. The interpolation of cyclic

deterioration properties is not presented in the figure, but is based on design

base shear just as the monotonic backbone properties have been.

Figure E-9 Example interpolation of nonlinear monotonic

backbone properties for the second story of the 4-

story, high gravity archetype (P-∆ effects not included)

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20

Sto

ry

Sh

ea

r (

kip

s)

Displacement (in)

Low Base Shear (factor of 1.0)

High Base Shear (factor of

1.32)

Interpolated Intermediate

Model (base shear factor of

1.16)

50% Draft


E.4.4 Non-Simulated Collapse Modes

Collapse is defined in a number of different ways for this study. For IDA, a

building is considered to collapse when the maximum interstory drift ratio

begins to increase rapidly, without any significant increase in ground motion

intensity (side-sway collapse). However, two other forms of collapse are

considered in addition to sideway collapse: 1) Failure of the (unmodeled)

gravity system and 2) Loss of vertical load carrying capacity of the lateral

system, due to shear failure of a column and its subsequent loss of ability to

carry gravity loads.

Neither shear failure modes nor gravity system failure are simulated by the

simplified or OMF frame models, so these failure modes are assessed

through non-simulated methods. These failure modes are of interest because

both result in structural members no longer having the capacity to withstand

vertical loads, which can lead to building collapse.

No gravity systems are design or modeled in this study, but it is still

important to acknowledge the fact that collapse in real buildings can result

due to failure of gravity elements, even if the lateral system is still in tact.

Assessing non-simulated collapse due to failure of the gravity system is

achieved in this study by setting a threshold interstory drift, beyond which

the gravity system is assumed to fail. If the maximum interstory drift in any

story of a building exceeds that threshold, then the building is assumed to

collapse. Thresholds of 3% and 6% were used for assessing non-simulated

collapse due to failure of the gravity system. These thresholds were chosen to

represent the range in ductility in gravity-load bearing systems possible in

SDC B.

Design standards for OMF’s do not require capacity design, so, as a result,

transverse reinforcement may be inadequate for carrying loads associated

with plastic hinging of the columns, resulting in brittle shear failure. This

specific type of brittle failure only applies to SDC B reinforced concrete

columns, but it is still relevant to include when we are trying to use OMF’s to

represent a SDC B lateral systems in general, because several other systems

with low R-factors are prone to brittle failure as well (joint shear failure and

weld failure in steel frames for example).

Column shear failure has been shown to depend on a combination of

displacement demand and shear force demand (Aslani 2005, and Elwood

2004). Therefore, the second non-simulated collapse mode, loss of vertical

load carrying capacity, is also assessed using interstory drift thresholds.

However, the drift thresholds are story specific, because the expected column

drift for which shear failure occurs depends on multiple parameters such as

41.86% Draft


column dimensions, axial load ratio, and reinforcement detailing. Using

those parameters, Aslani (2005) and Elwood (2004) have developed

empirical methods for predicting the probabilities of shear failure and

subsequent loss of gravity-load bearing capacity in reinforced concrete

columns.

In this study, column drifts corresponding to a 50% probability of loss

vertical load carrying capacity are computed according to the methods of

Aslani (2005) and Elwood (2004) and interpreted as non-simulated collapse

related to column shear failure. These column drifts are then mapped to total

interstory drifts using results from static pushover analysis, accounting for

drift contributions from column, beam, and joint rotations. Collapse occurs if

the drift in any column exceeds the collapse interstory drift threshold. For

example, the loss of vertical load carrying capacity drift threshold for a

second story interior column in the 4-story, high gravity, archetype is 1.80%,

but the interstory drift threshold for non-simulated collapse for the second

story is taken to be 2.35%, due to the portion of the drift resulting from beam

and joint rotations.

The adjusted collapse margin ratio for each archetype varies significantly

with varying methods of assessing non-simulated collapse, however, the

relative improvement gained from designing for accidental torsion in this

study is mostly independent of which, if any, non-simulated collapse

mechanism is implemented. Therefore, all of the results figures combine the

results from each of the non-simulated collapse modes considered in addition

to the results obtained without non-simulated collapse, unless otherwise

specified. Complete results are provided in the subsequent section, E.5.

E.5 Sensitivity of Collapse Risk Assessments to Designing for Accidental Torsion

Results of the assessments, in terms of the change in collapse risk due to

designing SDC B buildings with and without accidental torsion, and the

absolute collapse risk (ACMR or probability of collapse), are presented in

this section. The following figures and paragraphs describe the main trends

observed in this study. These trends include:

Trends specifically relevant to the scope of the study:

• Torsionally flexible buildings benefit more from being designed for

accidental torsion than torsionally stiff buildings. As a result, the

relative frame spacing parameter (S/L) is an excellent predictor of

the effectiveness of designing accidental torsion for all building

50% Draft


types studied. In addition, torsionally flexible buildings perform

much worse overall, with greater absolute collapse probabilities.

• Inherent torsion had little or no impact on the effectiveness of

designing for accidental torsion, but it does lower absolute collapse

capacity significantly.

• The torsional irregularity ratio that is computed in Table 12.3-1 of

ASCE 7 is a good, but sometimes conservative predictor, of the

effectiveness of designing for accidental torsion. Collapse capacity

decreases as torsional irregularity ratio increases.

Other Trends:

• Buildings with intermediate torsional flexibility perform moderately

better than torsionally stiff buildings and much better than torsionally

flexible buildings, when measured in terms of absolute collapse risk.

• Lightweight buildings perform better than heavy buildings.

• Short buildings perform better than tall buildings for the range of

building heights used in the study.

E.5.1 Trends Specifically Relevant to the Scope of this Study

E.5.1.1 Torsional Stiffness Measured by Relative Frame Spacing

Designing for accidental torsion in archetypes with moderate to high

torsional stiffness (0.5≤S/L≤1.0) makes very little difference in their collapse

performance, as shown in Figure E-10. The relative improvements in ACMR

due to including accidental torsion in the design are less than 10% for the

majority of cases and 2.1% on average. In this range, improvements due to

designing for accidental torsion are fairly constant regardless of the relative

frame spacing. In contrast, torsionally flexible (i.e. S/L<0.5) archetypes see

significant improvements in collapse capacity when they are designed for

accidental torsion. Each line in

Figure E-13 represents average relative improvement of collapse capacity

for each archetypical building as relative frame spacing (S/L) is varied. The

trend is virtually flat when 0.5≤S/L≤1.0, but steep (showing increasingly

pernicious consequences from not designing for accidental torsion) when

S/L<0.5.

The influence of relative frame spacing (S/L) on the effectiveness of design

accidental torsion was quantitatively compared to the influence of the other

design/modeling parameters by means if a binary regression tree, Figure E-

0.25 0.3 0.35 0.4 0.45 0.5 0.75 1-0.05

0

0.05

0.1

Relative Frame Spacing (S/L)

Re

lati

ve

Im

pro

ve

me

nt

of

AC

MR

Inherent Tors

Mean of Control Cases

41.86% Draft


10. The other parameters are: plan configuration, inherent torsion (yes/no),

number of stories, gravity load level, and center of mass offset. The

following binary regression tree was obtained by splitting the data into

optimal binary categories such that the total variance of the categorized data

was minimized.

Figure E-10 Binary regression tree for relative improvements of

ACMR

The regression tree of Figure E-10 shows that the most significant portion

of the variance in the data is captured by relative frame spacing (S/L). When

the results are categorized as (S/L)>0.425 and (S/L)<0.425, the expected

values of ACMR improvements for the two categories are 2.7% and 10.7%,

respectively.

Cross-validated error estimates are computed for each split of the binary

regression tree in Figure E-10 and are shown in Figure E-11. The results

show that only the first split, which is based on relative frame spacing (S/L),

is appropriate for this data set, because any additional splits do not lower the

error total error from cross-validated estimation. In other words, relative

frame spacing (S/L) is the single most influential factor for predicting the

effect of design accidental torsion, for this particular data set. The pruned

regression tree for relative improvement of ACMR is presented in Figure E-

12.

50% Draft


Figure E-11 Relative error obtained thru cross-validation vs.

number of splits for the binary regression tree of

relative ACMR improvement due to design accidental

torsion. Only the first split decreased the error

significantly.

41.86% Draft


Figure E-12 Pruned binary regression tree for relative

improvements of ACMR

Figure E-13 Average relative improvement of ACMR for subgroups

of buildings due to design accidental torsion, as a

function of relative frame spacing

A modified version of a statistical method called change-point analysis was

used to try to pin-point the exact location where the influence of design

accidental torsion on collapse capacity begins to really kick in. By visual

inspection of Figure E-13, it can be observed that the slope of the lines must

change somewhere in the range 0.25<(S/L)<0.5, but data within that range

has only been obtained for the 1-story high gravity case.

The essence of change-point analysis is to detect jumps in a data set by

fitting local polynomial regressions to data only on one side of each point

and then only to the other side. A jump in the data is indicated when the

squared difference of the value of the two local regression lines at a point

(one local polynomial fitted to the data on each side of the point) is large in

comparison to other points. However, we are concerned with a sudden

change in slope, not an actual jump, so the method has been refined to

compare slopes of local polynomials rather than values (hereafter referred to

as ‘change-slope’ analysis).

0.25 0.3 0.35 0.4 0.45 0.5 0.75 1-0.05

0

0.05

0.1

0.15

0.2


Re

lati

ve

Im

pro

ve

me

nt

of

AC

MR

10-Story High Grav

4-Story Low Grav

4-Story High Grav

1-Story Low Grav

1-Story High Grav

I-shape

Inherent Tors


50% Draft


0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Ch

an

ge

-Slo

pe

Va

lue

fro

m L

oc

al

Po

lyn

om

ial

An

aly

sis

Ninetieth Quantile

Median

Tenth Quantile

In local polynomial regression, the two main parameters that control the way

the regression is fit are the portion of neighboring data points used (alpha)

and the degree of the polynomial. Alpha values of 0.5-0.9 in increments of

0.05 with polynomial degree equal to 1.0 were used for performing the

change-slope analysis (9 combinations).

Due to the relatively large dispersion of the data, fitting the left-side local

polynomial to at small relative frame spacing was difficult. This difficulty

has been overcome by simulating data and doing multiple iterations. Data

was simulated using local polynomials of degree one with alpha values

ranging from 0.6 to 9 in increments of 0.05 (7 combinations). For each

combination of polynomial degree and alpha, three hundred data points were

simulated from the original data, and a change-slope analysis was performed

at each point. This process was repeated 100 times for each combination of

data simulation parameters and change-slope parameters for a total number

of 9x7=63 combinations. For each of the 63 combinations, the median

change-slope values were retained at each point.

Due to the sensitivity of the change slope analysis to data variance at small

values of relative frame spacing (S/L), some of the change-slope analysis

gave bogus results at the left side. This problem was remedied by looking at

the range 0.325<(S/L)<0.7 and rejecting any analysis that showed a

maximum change-slope at (S/L)=0.325. From the remaining analyses, an

envelope of change slope values was created and is shown in

Figure E-14.

41.86% Draft


0.2 0.4 0.6 0.8 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25


Rel

ativ

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S

cale

d C

hang

e−P

oint

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)

DataLocal Poly FitMean−stdErrorMean+stdErrorChange−Point Result

Figure E-14 Envelope of successful change-slope analyses (57 of

63 analyses were considered successful)

The envelope above indicates that the location of the most sudden change in

slope likely occurs at a relative frame spacing (S/L) of approximately 0.45.

In addition to the envelope of change-slope analyses results, a t-distribution

of peak points of each successful analysis was formed, for which the 90%

confidence range was 0.36<(S/L)<0.53 with a mean of 0.45. A sample of the

change-slope analysis results from 100 iterations of a single combination of

parameter values is shown in Figure E-15.

Figure E-15 Example of a single change-slope analysis result

obtained from 80 iterations with a local polynomial of

degree 1 and alpha=0.7 for simulating data and

alpha=0.65 for doing change-slope analysis. The

circles are the original data. The red line and dashed

black lines represent the local polynomial parameters

for simulating the data, and the blue line is the scaled

change-slope analysis result.

50% Draft


Results also show that torsionally flexible buildings (relative frame spacing

(S/L) <0.5) have much lower absolute collapse capacities than their

torsionally stiff counterparts. Figure E-16 shows that collapse capacity

increases as relative frame spacing increases for every archetype group when

relative frame spacing (S/L) is less than 0.5, but plateaus when S/L≥0.5.

Figure E-16 Average absolute ACMR vs. relative frame spacing.

Non-simulated collapse modes not considered. Thin

lines represent cases where accidental torsion is

considered in design

A pruned binary regression tree has also been constructed for absolute

ACMR and is presented in Figure E-17. Binary regression analysis shows

that gravity load level, relative frame spacing (S/L), and number of stories

are the most influential factors for collapse capacity, in that order. For the

case of absolute collapse capacity, relative frame spacing is still a significant

contributor, but it is not the only important factor, nor is it the most

important.

It can also be observed that none of the splits on the pruned regression tree

for ACMR are for design accidental torsion, which indicates that its

contribution to collapse capacity in this study is much smaller than the

contributions from gravity load, relative frame spacing, and number of

stories. For an ideal case in which building code requirements and the design

0.25 0.3 0.35 0.4 0.45 0.5 0.75 11

1.5

2

2.5

3

3.5


AC

MR

10-Story High Grav

4-Story Low Grav

4-Story High Grav

1-Story Low Grav

1-Story High Grav

I-shape

Inherent Tors

41.86% Draft


0.1

0.12

0.14

0.16

0.18

0.2

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S/L=0.5

S/L=0.25

of our archetypes are both perfect, design accidental torsion would be the

single most important factor for predicting collapse capacity for two reasons:

1) code requirements are supposed to make collapse capacity independent of

building factors such as weight and height and 2) the design accidental

torsion requirement is supposed to make building collapse capacity

independent of torsional flexibility.

Figure E-17 Pruned binary regression tree for absolute ACMR (non-

simulated collapse modes omitted)

E.5.1.2 Effects of Inherent Torsion

The presence of inherent torsion in an archetype building did not

significantly influence the effect of designing for accidental torsion in this

study.

Figure E-18 shows the average relative improvements of collapse capacity

for the 1-story high gravity archetypes with relative frame spacing (S/L) of

0.5 and 0.25. Two of the archetypes shown have inherent torsion (high

levels of the torsional irregularity ratio) and the other two are regular. The

slopes of the lines are virtually flat, indicating that inherent torsion is not a

determining factor for the effect of designing for accidental torsion. The

torsional irregularity ratio is discussed in more detail in the next section.

50% Draft


Figure E-18 Effect of inherent torsion on collapse capacity for the

1-story high gravity archetype

However, it is important to note that the absolute collapse capacity of

buildings with inherent torsion is much lower than their symmetric

counterparts. Note that in Figure E-16, all blue lines represent 1-story high

gravity archetypes. In absolute terms, the collapse capacities of the

symmetric archetypes (labeled ‘1-Story High Grav’ and ‘I-shape’) are much

higher, in an absolute sense, than the collapse capacity of the 1-story high

gravity archetype that has inherent torsion.

E.5.1.3 Torsional Irregularity Ratio

Since relative frame spacing (S/L) is not a practical metric for categorizing

buildings with more than two frame lines in each orthogonal direction, the

results have been recast in terms of torsional irregularity ratio, which is

computed according to Table 12.3-1 of ASCE 7-10, by completing the

following steps:

• Apply a lateral load which is offset from the center of mass perpendicular to

the direction of loading by a distance 5% of the buildings longest dimension

perpendicular to the direction of loading

• Take a ratio of the largest displacement parallel to the applied load at any

point in the plan of the building to the average displacement parallel to the

applied load

• Repeat with 5% offset in the opposite direction

• Repeat for each main orthogonal direction

• Take the largest of the computed displacement ratios

Contributions to torsional irregularity can come from either inherent torsion

or torsional flexibility (torsional flexibility contributes to torsional

41.86% Draft


irregularity because a 5% offset for accidental torsion must be considered

when the torsional irregularity ratio is computed). For symmetric buildings

(i.e. those without inherent torsion), the torsional irregularity ratio is a good

predictor of the effectiveness of designing for accidental torsion, as shown in

Figure E-19, because it is directly related to relative frame spacing.

Although buildings with inherent torsion have higher torsional irregularity

ratios than their symmetric counterparts that have the same relative frame

spacing (S/L), they see similar improvements from designing for accidental

torsion. Therefore, using torsional irregularity as an indicator for the

importance of accidental torsion in design penalizes buildings with inherent

torsion more readily than those without inherent torsion. The conservatism

of using torsional irregularity ratio to predict the benefits of design accidental

torsion in buildings with inherent torsion can be observed by examining

Figure E-13 and Figure E-19. Note that in

Figure E-13, where relative frame spacing is on the x-axis, all of the blue

lines follow the same basic trend. However, in Figure E-19 below, the dotted

blue line that represents the 1-story high gravity case with inherent torsion is

far below the other blue lines. Using the torsional irregularity ratio as a

trigger to require accidental torsion therefore identifies buildings with

inherent torsion as candidates for being designed with accidental torsion

more readily than symmetric buildings, despite the observation that the

degree of inherent torsion is not highly related to the importance of

accidental torsion.

0.25 0.3 0.35 0.4 0.45 0.5 0.75 1-0.05

0

0.05


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0.25 0.3 0.35 0.4 0.45 0.5 0.75 1-0.05


50% Draft


Figure E-19 Relative improvement of collapse capacity due to

designing for accidental torsion vs. torsional

irregularity ratio

Binary regression analysis also confirms the observation that torsional

irregularity ratio is strong factor for relative ACMR improvement due to

design accidental torsion requirements. When torsional irregularity ratio is

substituted for the combination relative frame spacing, plan configuration,

and inherent torsion, the result is very similar to that obtained using relative

wall spacing as a parameter.

1 1.5 2 2.5 3 3.5 4 -0.05

0

0.05

0.1

0.15

0.2

Torsional Irregularity Ratio

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10-Story High Grav

4-Story Low Grav

4-Story High Grav

1-Story Low Grav

1-Story High Grav

I-shape

Inherent Tors

41.86% Draft


1 1.5 2 2.5 3 3.5 4 1

1.5

2

2.5

3

3.5

Torsional Irregularity Ratio

AC

MR

10-Story High Grav

4-Story Low Grav

4-Story High Grav

1-Story Low Grav

1-Story High Grav

I-shape

Inherent Tors

Figure E-20 Pruned binary regression tree for relative

improvements of ACMR, using torsional irregularity

ratio

An additional change-slope analysis of the results from using torsional

irregularity ratio was not done, because the torsional irregularity ratios can be

computed directly from the relative frame spacing values for the 1-story high

gravity building results. When t-test results from section E5.1.1 are

converted to torsional irregularity ratios, the 90% confidence range for the

‘kink’ in the data is 1.29<torsional irregularity ratio<1.58, with an expected

value of 1.40.

Torsional irregularity ratio is also good predictor for absolute collapse

capacity, as can be seen in Figure E-21. Note that the thick dotted blue line

that represents the 1-story high gravity building with inherent torsion

coincides with the thick blue lines representing the symmetric 1-story high

gravity buildings, which can be contrasted to the results that are based on

relative wall spacing (S/L) in Figure E-16. Therefore, torsional irregularity

ratio is an equal or better predictor than relative wall spacing for absolute

collapse capacity.

Figure E-21 Absolute capacity vs. torsional irregularity ratio (non-

simulated collapse modes omitted)

50% Draft


It should be noted that, although Figure E-21 shows a strong correlation

between torsional irregularity ratio and collapse capacity, there are other

significant factors in this study that affect collapse capacity as well. The

pruned regression tree of ACMR in Figure E-22 shows that gravity load level

and number of stories are also important factors, just like when relative frame

spacing was used.

Figure E-22 Pruned binary regression tree of ACMR using torsional

irregularity ratio (non-simulated collapse modes

omitted)

E.5.2 Other Trends

E.5.2.1 Better Performance in Buildings with Intermediate Torsional Flexibility

An unexpected trend observed in Figure E-16 is a slight increase in collapse

capacity when buildings transition from being torsionally stiff (S/L~1.0) to

moderately torsionally stiff (S/L~0.5), even though the structure’s torsional

resistance is decreasing. It seems logical that greater torsional flexibility

would lead to greater deformations and, therefore, lower collapse capacity.

This observation is true in general, but there is a minor increase in collapse

capacity when torsional stiffness reduces from very stiff to moderately stiff.

This slight increase occurs because as the fundamental torsional period

increases with increased torsional flexibility, the building falls into a

frequency range for which the earthquake records have much less spectral

energy. This effect partly counterbalances the larger displacements that occur

41.86% Draft


due to the increased torsional flexibility. As an illustrative example, the first

seven modal periods for the 3D 4-story high gravity archetype are shown

below in Table E-7 (torsional periods are highlighted in grey) along with a

spectral acceleration plot of the FEMA P695 far-field ground motion set

(Figure E-23). Note that spectral acceleration is highest for short periods and

decreases steadily as the fundamental period increases beyond 0.5 seconds,

which means that buildings with short torsional periods (i.e. torsionally stiff

buildings), will experience the greatest torsional accelerations and torsional

flexible buildings experience relatively less demand.

Table ETable ETable ETable E----7777 Summary of Summary of Summary of Summary of MMMModal odal odal odal PPPPeriods for the 4eriods for the 4eriods for the 4eriods for the 4----SSSStory tory tory tory HHHHigh igh igh igh GGGGravity ravity ravity ravity

AAAArchetyperchetyperchetyperchetype

4-story High Gravity

Mode

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

1 2.34 lat 2.34 lat 2.70 tors 6.78 tors

2 2.34 lat 2.34 lat 2.34 lat 2.34 lat

3 1.30 tors 1.75 tors 2.34 lat 2.34 lat

4 0.83 lat 0.83 lat 0.96 tors 2.21 tors

5 0.83 lat 0.83 lat 0.83 lat 1.32 tors

6 0.51 lat 0.63 tors 0.83 lat 0.91 tors

7 0.51 lat 0.51 lat 0.59 tors 0.83 lat

50% Draft


Figure E-23 Acceleration spectrum of the FEMA P695 ground

motion set (FEMA P695 Report)

E.5.2.2 Lightweight vs. Heavy Buildings

Gravity load level is also a major contributor to building collapse

performance as is illustrated by the regression trees in Figure E-17 and

Figure E-22. Figure E-24 also illustrates the effects of gravity loads on

collapse performance for the one and four story archetypes; heavier buildings

(red) tended to perform worse than lightweight buildings (black) in this

study. However, the relative improvement from designing for accidental

torsion is fairly similar between high and low gravity archetypes (note that

the red line trends are similar to black line trends in Figure E-25 and also the

binary regression tree in Figure E-10).

A major contributor to the difference in absolute collapse capacity between

the high and low gravity archetypes is the fact that transverse reinforcement

spacing was typically controlled by the maximum allowable spacing, rather

than by design loads. Since the high gravity frames have larger sections,

their maximum allowable spacing for transverse reinforcement is also larger,

lead to less ductile beam and column elements and overall worse

performance.

41.86% Draft


0.25 0.3 0.35 0.4 0.45 0.5 0.75 12


4-Story Low Grav

1-Story Low Grav

4-Story High Grav

1-Story High Grav

Figure E-24 Influence of gravity load level and building height on

absolute collapse capacity. Non-simulated collapse

modes not considered.

Figure E-25 Influence of gravity load level and building height on

the effect of design accidental torsion on collapse

capacity.

E.5.2.3 Effect of Building Height on Collapse Performance

Taller buildings tend to perform worse than shorter buildings in this study.

The tallest building analyzed is 10 stories (132 ft.), and its absolute collapse

capacity is the lowest of all the archetypes (see Figure E-16 and Figure E-

21). Additionally, Figure E-24 shows that the 4-story buildings (dashed

lines) performed worse than their 1-story counterparts (solid lines). After

about 10 stories, equation 12.8.5 of ASCE 7-10, which defines a minimum

value of 0.01g for the base shear coefficient Cs, is triggered, reducing the

effect of building height in the design. One significant contributor to the

greater performance of the shorter buildings, which is specific the building

types in this study, but not necessarily in general, is the of the OMF design

requirements for maximum transverse reinforcement spacing. Due to

maximum shear spacing requirements, the shorter buildings in this study tend

to be more ductile. For example, the design shear in the columns of the 1-

story OMF’s is so low that transverse reinforcement is almost not even

needed for resisting shear forces. However, since the design of transverse

0.25 0.3 0.35 0.4 0.45 0.5 0.75 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


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4-Story Low Grav

1-Story Low Grav

4-Story High Grav

1-Story High Grav

50% Draft


reinforcement is controlled by maxim spacing requirements, the columns are

‘over-designed’ for shear forces. The transverse reinforcement spacing is so

low, in fact, that the columns exceed capacity design standards, which makes

them very ductile, even though capacity design is not required for OMF’s.

E.6 Conclusions and Recommendations

E.6.1 Revision of Section 12.8.4.2 in ASCE 7

According to the results of this study, the single most significant predictor for

the improvement of building collapse capacity due to designing for

accidental torsion is relative frame spacing (S/L). Buildings with relative

frame spacing (S/L) greater than 0.45 tend see very little improvement in

their collapse capacities when designed for accidental torsion (see

Figure E-13,Figure E-12, and Figure E-14); therefore, we recommend that

such buildings need not adhere to the accidental torsion requirement of

ASCE 7-10. However, computing an ‘equivalent S/L relative frame spacing’

for buildings with multiple frame lines is burdensome and prone to

misinterpretation.

As a result, an alternative method, using torsional irregularity ratio, is

suggested, because ASCE 7-10 already requires that quantity to be

computed. As shown above, the torsional irregularity ratio is a good

predictor of relative improvement of collapse capacity (see section E.5.1.3

and Figure E-19), and buildings with torsional irregularity ratio less than 1.4

gain very little, in terms of collapse capacity, from design accidental torsion.

A torsional irregularity ratio of 1.4 is also the cut-off or torsional irregularity

type 1b, therefore, we recommend that buildings designed for SDC B,

which do not have type 1b torsional irregularity, need not adhere to the

accidental torsion requirement of section 12.8.4.2 of ASCE 7-10. Since

our results indicate that torsional flexibility, rather than inherent torsion, is

critical in determining the effect of designing for accidental torsion on

collapse capacity (see

Figure E-18), this recommendation conservatively affects buildings with

inherent torsion. However, buildings with inherent torsion perform more

poorly in general than their symmetric counterparts in terms of absolute

collapse capacity (see Figure E-16), such that higher levels of conservatism

in design may be appropriate.

0.25 0.3 0.35 0.4 0.45 0.5 0.75 1-0.05

0


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41.86% Draft


E.6.2 Future Research

It is desirable for buildings of all levels of torsional irregularity due to

torsional flexibility or inherent torsion to have similar collapse capacities; if

this were the case, the lines in Figure E-21 would be flat, indicating that

collapse capacity is independent of torsional irregularity ratio. However, as

shown in Figure E-21 the collapse capacity declines rapidly after the

torsional irregularity ratio increases past 1.4 and has decreased 20-50% by

the time the torsional irregularity ratio is approximately 2.3. Therefore, an

amplification factor for torsionally flexible buildings in SDC B that would

increase their collapse capacities to a level comparable to the collapse level

when torsional irregularity ratio is less than 1.4 is suggested for future

research.

Additionally, the authors recommend a similar future study to examine the

effects of design accidental torsion in SDC D. Such a study would have at

least two benefits:

• A better knowledge of whether the torsion amplification factor, Ax,

in ASCE 7-10 is accomplishing its intended purpose of making

seismic collapse capacity independent of the level of torsional

irregularity or flexibility.

• Possible elimination of the accidental torsion design requirement for

buildings in high seismic regions that are not categorized as having

torsional irregularity type 1a or 1b.

50% Draft


E.7 Exhaustive Results Summary

E.7.1 All Archetype Collapse Results

Table ETable ETable ETable E----8888 Collapse Results for the Collapse Results for the Collapse Results for the Collapse Results for the 1111----Story, Story, Story, Story, Low Low Low Low GravityGravityGravityGravity, , , ,

Symmetric Symmetric Symmetric Symmetric ArchetypesArchetypesArchetypesArchetypes

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3011100 1 0.37 3.43 0.03 0.37 3.40 0.03 0.20 1.79 0.18

3011200 0.75 0.37 3.43 0.03 0.37 3.40 0.03 0.20 1.79 0.18

3011300 0.5 0.37 3.43 0.03 0.37 3.40 0.03 0.20 1.79 0.18

3011400 0.25 0.37 3.43 0.03 0.37 3.40 0.03 0.20 1.79 0.18

3011110 1 0.38 3.46 0.03 1% 0.37 3.41 0.03 0% 0.20 1.86 0.17 4%

3011210 0.75 0.38 3.49 0.03 2% 0.38 3.44 0.03 1% 0.20 1.88 0.17 5%

3011310 0.5 0.39 3.54 0.03 3% 0.38 3.45 0.03 2% 0.21 1.93 0.16 7%

3011410 0.25 0.38 3.52 0.03 2% 0.37 3.44 0.03 1% 0.21 1.92 0.16 7%

3011100 1 0.37 3.43 0.03 0.37 3.40 0.03 0.20 1.79 0.18

3011200 0.75 0.38 3.46 0.03 0.37 3.42 0.03 0.20 1.79 0.18

3011300 0.5 0.37 3.44 0.03 0.37 3.41 0.03 0.20 1.79 0.18

3011400 0.25 0.31 2.82 0.06 0.29 2.68 0.06 0.19 1.75 0.19

3011110 1 0.38 3.45 0.03 0% 0.37 3.41 0.03 0% 0.20 1.86 0.17 4%

3011210 0.75 0.38 3.48 0.03 1% 0.37 3.43 0.03 0% 0.20 1.88 0.17 5%

3011310 0.5 0.38 3.52 0.03 2% 0.38 3.45 0.03 1% 0.21 1.93 0.16 7%

3011410 0.25 0.33 2.99 0.05 6% 0.30 2.71 0.06 1% 0.21 1.91 0.16 9%

3011101 1 0.33 3.05 0.04 0.32 2.96 0.05 0.16 1.43 0.29

3011201 0.75 0.34 3.14 0.04 0.33 3.06 0.04 0.15 1.39 0.31

3011301 0.5 0.36 3.34 0.03 0.34 3.15 0.04 0.14 1.29 0.35

3011401 0.25 0.27 2.52 0.08 0.23 2.14 0.12 0.14 1.25 0.36

3011111 1 0.34 3.10 0.04 1% 0.33 3.02 0.04 2% 0.17 1.52 0.26 7%

3011211 0.75 0.35 3.18 0.04 1% 0.33 3.02 0.04 -1% 0.15 1.41 0.30 1%

3011311 0.5 0.37 3.38 0.03 1% 0.36 3.28 0.03 4% 0.14 1.31 0.34 1%

3011411 0.25 0.30 2.74 0.06 9% 0.26 2.41 0.09 13% 0.13 1.21 0.39 -4%

3011102 1 0.33 3.04 0.04 0.33 3.00 0.05 0.15 1.41 0.30

3011202 0.75 0.35 3.17 0.04 0.33 3.02 0.04 0.15 1.38 0.31

3011302 0.5 0.36 3.30 0.03 0.33 3.03 0.04 0.12 1.13 0.42

3011402 0.25 0.26 2.40 0.09 0.20 1.86 0.17 0.10 0.96 0.52

3011112 1 0.33 2.99 0.05 -2% 0.32 2.94 0.05 -2% 0.15 1.36 0.32 -4%

3011212 0.75 0.35 3.21 0.04 1% 0.34 3.08 0.04 2% 0.16 1.44 0.29 5%

3011312 0.5 0.37 3.38 0.03 2% 0.34 3.10 0.04 2% 0.12 1.06 0.47 -7%

3011412 0.25 0.30 2.71 0.06 13% 0.23 2.10 0.13 13% 0.11 1.01 0.49 5%

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41.86% Draft


*Total logarithmic dispersion (b) is assumed to be 0.65 for computing

P(Collapse|MCE).

**The natural eccentricity is zero, but 2small amounts of torsion are

introduced due to the nature of the simplified frames.

2 Unless the rotational degrees of freedom of floor diaphragms are fixed in

the OpenSees models, small amounts of torsion are introduced in the 3D

models due to geometric nonlinearities. This effect occurs due to the

configuration of the simplified models. When a frame or wall resists a lateral

load, it must be stabilized in the out-of-plane direction to prevent it from

twisting. Taking framed systems as an example, when out-of-plane

deflections are present, compression forces in the columns are destabilizing

and push the frame farther out of plane. If there is any lateral load acting

along the plane of the frame, there will be more compression in the columns

of one side than the other, cause varying destabilizing forces at each end,

which results in a moment; thus, ‘accidental torsion’ is introduced. A similar

phenomenon occurs from the braces as well; tension braces tend to pull one

end of the top of the frame back to its vertical in-plane position, while

compression braces tend to push the top of frames at the opposite end farther

out plane. The extent to which geometric nonlinearities contribute to natural

accidental torsion in the simplified models can be observed in the 1-story

rows of Table E-8 and Table E-9 in which the perfectly symmetric 1-story

archetypes have been analyzed with and without the rigid diaphragm being

allowed to rotate. For cases where S/L is greater than or equal to 0.5 (i.e. the

building was not extremely torsionally flexible), these geometric

nonlinearities contribute very little to natural accidental torsion.

50% Draft


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3012100 1 0.28 2.85 0.05 0.28 2.80 0.06 0.21 2.07 0.13

3012200 0.75 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012300 0.5 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012400 0.25 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012110 1 0.29 2.94 0.05 3% 0.29 2.88 0.05 3% 0.21 2.10 0.13 2%

3012210 0.75 0.30 2.97 0.05 4% 0.29 2.94 0.05 5% 0.21 2.16 0.12 6%

3012310 0.5 0.31 3.08 0.04 8% 0.30 3.00 0.05 7% 0.22 2.17 0.12 7%

3012410 0.25 0.32 3.16 0.04 11% 0.31 3.12 0.04 11% 0.21 2.15 0.12 6%

3012100 1 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012200 0.75 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012300 0.5 0.28 2.85 0.05 0.28 2.80 0.06 0.20 2.03 0.14

3012500 0.45 0.28 2.85 0.05 0.28 2.80 0.06 0.16 1.58 0.24

3012600 0.4 0.28 2.84 0.05 0.28 2.78 0.06 0.16 1.58 0.24

3012700 0.35 0.28 2.79 0.06 0.27 2.73 0.06 0.16 1.56 0.25

3012800 0.3 0.26 2.65 0.07 0.25 2.46 0.08 0.15 1.54 0.25

3012400 0.25 0.23 2.33 0.10 0.22 2.20 0.11 0.18 1.84 0.18

3012110 1 0.29 2.93 0.05 3% 0.29 2.88 0.05 3% 0.21 2.13 0.12 5%

3012210 0.75 0.29 2.96 0.05 4% 0.29 2.94 0.05 5% 0.21 2.16 0.12 6%

3012310 0.5 0.31 3.07 0.04 8% 0.30 3.00 0.05 7% 0.22 2.16 0.12 6%

3012510 0.45 0.31 3.07 0.04 8% 0.30 3.01 0.05 7% 0.15 1.55 0.25 -2%

3012610 0.4 0.30 3.04 0.04 7% 0.30 2.98 0.05 7% 0.16 1.62 0.23 3%

3012710 0.35 0.30 3.06 0.04 10% 0.29 2.90 0.05 6% 0.16 1.64 0.22 5%

3012810 0.3 0.29 2.95 0.05 12% 0.28 2.81 0.06 14% 0.16 1.64 0.22 6%

3012410 0.25 0.27 2.71 0.06 16% 0.26 2.60 0.07 18% 0.22 2.17 0.12 18%

3012101 1 0.25 2.47 0.08 0.24 2.45 0.08 0.16 1.63 0.23

3012201 0.75 0.25 2.56 0.07 0.25 2.53 0.08 0.15 1.52 0.26

3012301 0.5 0.28 2.77 0.06 0.27 2.70 0.06 0.14 1.43 0.29

3012501 0.45 0.27 2.73 0.06 0.27 2.69 0.06 0.16 1.59 0.24

3012601 0.4 0.27 2.75 0.06 0.26 2.63 0.07 0.15 1.54 0.25

3012701 0.35 0.26 2.66 0.07 0.25 2.55 0.07 0.15 1.46 0.28

3012801 0.3 0.25 2.50 0.08 0.21 2.16 0.12 0.15 1.47 0.28

3012401 0.25 0.21 2.10 0.13 0.17 1.70 0.21 0.12 1.25 0.37

3012111 1 0.26 2.63 0.07 7% 0.26 2.58 0.07 5% 0.17 1.67 0.22 2%

3012211 0.75 0.27 2.72 0.06 6% 0.27 2.68 0.06 6% 0.16 1.60 0.23 6%

3012311 0.5 0.30 2.98 0.05 8% 0.28 2.84 0.05 5% 0.14 1.43 0.29 0%

3012511 0.45 0.30 3.04 0.04 12% 0.29 2.87 0.05 7% 0.14 1.43 0.29 -10%

3012611 0.4 0.30 3.00 0.05 9% 0.28 2.83 0.05 7% 0.17 1.66 0.22 8%

3012711 0.35 0.30 2.99 0.05 13% 0.26 2.60 0.07 2% 0.16 1.60 0.24 9%

3012811 0.3 0.28 2.81 0.06 12% 0.24 2.45 0.08 14% 0.16 1.57 0.24 7%

3012411 0.25 0.25 2.50 0.08 19% 0.20 1.99 0.15 17% 0.14 1.39 0.31 11%

3012102 1 0.24 2.42 0.09 0.24 2.40 0.09 0.15 1.47 0.28

3012202 0.75 0.26 2.65 0.07 0.25 2.56 0.07 0.14 1.42 0.29

3012302 0.5 0.27 2.76 0.06 0.26 2.62 0.07 0.13 1.26 0.36

3012402 0.25 0.20 2.01 0.14 0.15 1.52 0.26 0.11 1.05 0.47

3012112 1 0.25 2.55 0.07 5% 0.25 2.47 0.08 3% 0.14 1.46 0.28 -1%

3012212 0.75 0.28 2.78 0.06 5% 0.26 2.63 0.07 3% 0.15 1.52 0.26 6%

3012312 0.5 0.29 2.87 0.05 4% 0.27 2.71 0.06 3% 0.12 1.23 0.38 -2%

3012412 0.25 0.23 2.34 0.10 16% 0.18 1.76 0.19 16% 0.11 1.14 0.42 8%

Na

tura

l E

cce

ntr

icit

y (

Off

set

of

CM

)

ID

Sto

rie

s

Gra

vit

y

Re

lati

ve

Fra

me

Sp

aci

ng

(S

/L)

De

sig

n f

or

Acc

ide

nta

l T

ors

ion

No

no

nsi

mu

late

d

Co

lla

pse

Mo

de

s

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

6%

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

3.5

%

1 160 psf

No

0

Yes

No

**0

Yes

No

5%

Yes

No

10%

Yes

Table ETable ETable ETable E----9999 Collapse Results for the 1Collapse Results for the 1Collapse Results for the 1Collapse Results for the 1----Story, High Gravity, Story, High Gravity, Story, High Gravity, Story, High Gravity,

Symmetric ArchetypesSymmetric ArchetypesSymmetric ArchetypesSymmetric Archetypes

41.86% Draft


Me

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Me

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Me

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4012300 0.5 0.28 2.83 0.05 0.28 2.78 0.06 0.16 1.58 0.24

4012500 0.45 0.28 2.79 0.06 0.27 2.75 0.06 0.16 1.58 0.24

4012600 0.4 0.27 2.73 0.06 0.27 2.68 0.06 0.16 1.56 0.25

4012700 0.35 0.26 2.59 0.07 0.24 2.45 0.08 0.15 1.52 0.26

4012800 0.3 0.24 2.42 0.09 0.23 2.31 0.10 0.15 1.53 0.26

4012400 0.25 0.20 2.03 0.14 0.18 1.85 0.17 0.15 1.47 0.28

4012310 0.5 0.30 2.99 0.05 6% 0.29 2.96 0.05 6% 0.16 1.61 0.23 2%

4012510 0.45 0.30 3.01 0.05 8% 0.29 2.91 0.05 6% 0.16 1.62 0.23 3%

4012610 0.4 0.30 2.98 0.05 9% 0.29 2.93 0.05 9% 0.16 1.63 0.23 4%

4012710 0.35 0.29 2.92 0.05 13% 0.28 2.80 0.06 14% 0.16 1.65 0.22 8%

4012810 0.3 0.28 2.80 0.06 16% 0.27 2.67 0.07 16% 0.17 1.67 0.21 10%

4012410 0.25 0.26 2.57 0.07 27% 0.23 2.29 0.10 24% 0.16 1.56 0.25 6%

4012301 0.5 0.28 2.76 0.06 0.26 2.62 0.07 0.11 1.08 0.46

4012501 0.45 0.27 2.71 0.06 0.25 2.55 0.08 0.11 1.08 0.45

4012601 0.4 0.26 2.60 0.07 0.24 2.40 0.09 0.11 1.11 0.44

4012701 0.35 0.25 2.48 0.08 0.22 2.22 0.11 0.11 1.12 0.43

4012801 0.3 0.22 2.17 0.12 0.19 1.95 0.15 0.11 1.10 0.44

4012401 0.25 0.17 1.72 0.20 0.15 1.52 0.26 0.11 1.11 0.44

4012311 0.5 0.30 2.98 0.05 8% 0.28 2.85 0.05 9% 0.11 1.11 0.43 3%

4012511 0.45 0.29 2.94 0.05 9% 0.26 2.65 0.07 4% 0.12 1.17 0.40 8%

4012611 0.4 0.28 2.85 0.05 10% 0.26 2.59 0.07 8% 0.12 1.22 0.38 10%

4012711 0.35 0.27 2.75 0.06 11% 0.25 2.55 0.07 15% 0.12 1.19 0.40 6%

4012811 0.3 0.26 2.58 0.07 19% 0.22 2.18 0.11 12% 0.12 1.16 0.41 5%

4012411 0.25 0.23 2.31 0.10 34% 0.18 1.83 0.18 21% 0.12 1.18 0.40 7%

ID

Sto

rie

s

Gra

vit

y

Re

lati

ve

Fra

me

Sp

aci

ng

(S

/L)

1 160 psf

No

**None

yes

No

5%

Yes

De

sig

n f

or

Acc

ide

nta

l T

ors

ion

Na

tura

l E

cce

ntr

icit

y (

Off

set

of

CM

)

No

no

nsi

mu

late

d

Co

lla

pse

Mo

de

s

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

6%

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

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Ed

ge

Ex

cee

ds

3%

Table ETable ETable ETable E----10101010 Collapse Results for the 1Collapse Results for the 1Collapse Results for the 1Collapse Results for the 1----Story, High Gravity, ‘IStory, High Gravity, ‘IStory, High Gravity, ‘IStory, High Gravity, ‘I----

ShapShapShapShaped’ Archetypesed’ Archetypesed’ Archetypesed’ Archetypes

50% Draft


Me

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5012300 0.5 0.22 2.25 0.11 0.22 2.18 0.12 0.11 1.06 0.46

5012400 0.25 0.16 1.64 0.22 0.15 1.51 0.26 0.10 0.98 0.51

5012310 0.5 0.24 2.41 0.09 7% 0.23 2.32 0.10 6% 0.11 1.11 0.44 5%

5012410 0.25 0.20 1.96 0.15 19% 0.17 1.68 0.21 11% 0.11 1.14 0.42 16%

5012301 0.5 0.21 2.13 0.12 0.20 2.01 0.14 0.10 1.02 0.49

5012401 0.25 0.16 1.56 0.25 0.14 1.42 0.30 0.09 0.85 0.60

5012311 0.5 0.21 2.12 0.12 0% 0.19 1.94 0.15 -3% 0.11 1.14 0.42 12%

5012411 0.25 0.18 1.85 0.17 19% 0.16 1.64 0.22 16% 0.09 0.94 0.54 10%

5012302 0.5 0.20 2.02 0.14 0.19 1.88 0.17 0.11 1.06 0.46

5012402 0.25 0.14 1.43 0.29 0.13 1.27 0.36 0.07 0.74 0.68

5012312 0.5 0.22 2.21 0.11 9% 0.21 2.13 0.12 13% 0.11 1.07 0.46 1%

5012412 0.25 0.18 1.83 0.18 28% 0.15 1.52 0.26 20% 0.09 0.89 0.57 20%

5012303 0.5 0.25 2.50 0.08 0.23 2.29 0.10 0.11 1.07 0.46

5012403 0.25 0.18 1.85 0.17 0.17 1.68 0.21 0.12 1.23 0.37

5012313 0.5 0.26 2.63 0.07 5% 0.25 2.46 0.08 7% 0.11 1.11 0.44 3%

5012413 0.25 0.21 2.11 0.13 14% 0.19 1.87 0.17 11% 0.12 1.25 0.36 2%

5012304 0.5 0.27 2.72 0.06 0.25 2.54 0.08 0.12 1.19 0.39

5012404 0.25 0.19 1.93 0.16 0.17 1.69 0.21 0.11 1.08 0.45

5012314 0.5 0.28 2.83 0.05 4% 0.25 2.53 0.08 0% 0.12 1.24 0.37 4%

5012414 0.25 0.22 2.25 0.11 17% 0.18 1.77 0.19 5% 0.12 1.19 0.40 10%

ID

Sto

rie

s

Gra

vit

y

Re

lati

ve

Fra

me

Sp

aci

ng

(S

/L)

De

sig

n f

or

Acc

ide

nta

l T

ors

ion

No

no

nsi

mu

late

d

Co

lla

pse

Mo

de

s

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

6%

No

nsi

mu

late

d

Co

lla

pse

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

3%

No

None

yes

Na

tura

l E

cce

ntr

icit

y (

Off

set

of

CM

)

No

-10%

yes

160 psf1

Yes

No

+10%

yes

No

-5%

Yes

No

+5%


Archetypes with Inherent TorsionArchetypes with Inherent TorsionArchetypes with Inherent TorsionArchetypes with Inherent Torsion

41.86% Draft


Me

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3041100 1 0.20 3.24 0.04 0.18 2.82 0.06 0.20 3.09 0.04 0.20 3.24 0.04 0.12 1.91 0.16

3041200 0.75 0.20 3.24 0.04 0.18 2.82 0.06 0.20 3.09 0.04 0.20 3.24 0.04 0.12 1.91 0.16

3041300 0.5 0.20 3.24 0.04 0.18 2.82 0.06 0.20 3.09 0.04 0.20 3.24 0.04 0.12 1.91 0.16

3041400 0.25 0.16 2.59 0.07 0.14 2.25 0.11 0.15 2.41 0.09 0.15 2.43 0.09 0.11 1.69 0.21

3041110 1 0.21 3.30 0.03 2% 0.18 2.87 0.05 2% 0.21 3.25 0.03 5% 0.21 3.30 0.03 2% 0.12 1.97 0.15 3%

3041210 0.75 0.21 3.30 0.03 2% 0.18 2.89 0.05 3% 0.21 3.26 0.03 5% 0.21 3.27 0.03 1% 0.12 1.98 0.15 3%

3041310 0.5 0.21 3.36 0.03 4% 0.18 2.93 0.05 4% 0.21 3.27 0.03 6% 0.21 3.28 0.03 1% 0.13 1.99 0.14 4%

3041410 0.25 0.18 2.82 0.06 9% 0.15 2.41 0.09 7% 0.16 2.52 0.08 4% 0.16 2.56 0.07 5% 0.11 1.77 0.19 5%

3041101 1 0.20 3.17 0.04 0.17 2.71 0.06 0.19 3.04 0.04 0.20 3.12 0.04 0.11 1.76 0.19

3041201 0.75 0.20 3.14 0.04 0.17 2.65 0.07 0.19 2.98 0.05 0.20 3.09 0.04 0.11 1.72 0.20

3041301 0.5 0.20 3.21 0.04 0.17 2.72 0.06 0.20 3.10 0.04 0.20 3.12 0.04 0.11 1.70 0.21

3041401 0.25 0.13 2.13 0.12 0.10 1.62 0.23 0.11 1.82 0.18 0.11 1.77 0.19 0.08 1.28 0.35

3041111 1 0.20 3.16 0.04 0% 0.17 2.66 0.07 -2% 0.19 3.05 0.04 1% 0.19 3.04 0.04 -3% 0.11 1.80 0.18 2%

3041211 0.75 0.20 3.22 0.04 2% 0.17 2.68 0.06 1% 0.20 3.12 0.04 5% 0.20 3.09 0.04 0% 0.11 1.76 0.19 2%

3041311 0.5 0.21 3.34 0.03 4% 0.18 2.77 0.06 2% 0.20 3.18 0.04 3% 0.21 3.30 0.03 6% 0.11 1.70 0.21 0%

3041411 0.25 0.16 2.47 0.08 16% 0.12 1.83 0.18 12% 0.12 1.96 0.15 8% 0.12 1.85 0.17 4% 0.08 1.30 0.34 2%

3041102 1 0.20 3.15 0.04 0.16 2.60 0.07 0.19 3.00 0.05 0.20 3.09 0.04 0.11 1.71 0.20

3041202 0.75 0.20 3.18 0.04 0.16 2.54 0.08 0.19 2.96 0.05 0.19 3.00 0.05 0.10 1.54 0.25

3041302 0.5 0.20 3.20 0.04 0.16 2.52 0.08 0.19 2.99 0.05 0.19 3.05 0.04 0.10 1.52 0.26

3041402 0.25 0.13 2.03 0.14 0.09 1.40 0.30 0.11 1.75 0.20 0.11 1.68 0.21 0.07 1.09 0.45

3041112 1 0.20 3.19 0.04 2% 0.16 2.56 0.07 -1% 0.19 3.02 0.04 1% 0.20 3.09 0.04 0% 0.11 1.70 0.21 -1%

3041212 0.75 0.20 3.19 0.04 0% 0.16 2.58 0.07 1% 0.19 3.04 0.04 3% 0.20 3.15 0.04 5% 0.10 1.59 0.24 3%

3041312 0.5 0.21 3.28 0.03 3% 0.16 2.55 0.07 1% 0.19 3.08 0.04 3% 0.19 3.04 0.04 0% 0.09 1.44 0.29 -5%

3041412 0.25 0.15 2.39 0.09 18% 0.10 1.63 0.23 17% 0.12 1.97 0.15 13% 0.11 1.81 0.18 7% 0.07 1.10 0.44 1%

4 100 psf

No

**0

No

10%

Yes

No

no

nsi

mu

late

d C

oll

ap

se

Mo

de

s

No

nsi

mu

late

d C

oll

ap

se

Occ

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No

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mu

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d C

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ap

se

Occ

urs

if

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at

Bu

ild

ing

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ge

Ex

cee

ds

3%

Yes

No

5%

Yes

No

nsi

mu

late

d C

oll

ap

se

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

6%

No

nsi

mu

late

d C

oll

ap

se

Occ

urs

if

IDR

at

Bu

ild

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Ed

ge

Ex

cee

ds

LVC

C (

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C

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or

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mn

ax

ial

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Na

tura

l E

cce

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y (

Off

set

of

CM

)

ID

Sto

rie

s

Gra

vit

y

Re

lati

ve

Fra

me

Sp

aci

ng

(S

/L)

De

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or

Acc

ide

nta

l T

ors

ion

Table ETable ETable ETable E----12121212 Collapse Results for the 4Collapse Results for the 4Collapse Results for the 4Collapse Results for the 4----Story, Low Gravity, Story, Low Gravity, Story, Low Gravity, Story, Low Gravity,


50% Draft


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3042100 1 0.17 2.70 0.06 0.14 2.13 0.12 0.16 2.48 0.08 0.17 2.70 0.06 0.12 1.83 0.18

3042200 0.75 0.17 2.70 0.06 0.14 2.13 0.12 0.16 2.50 0.08 0.17 2.70 0.06 0.12 1.83 0.18

3042300 0.5 0.17 2.70 0.06 0.14 2.13 0.12 0.16 2.50 0.08 0.17 2.70 0.06 0.12 1.83 0.18

3042400 0.25 0.15 2.39 0.09 0.13 1.98 0.15 0.14 2.17 0.12 0.14 2.23 0.11 0.11 1.68 0.21

3042110 1 0.18 2.78 0.06 3% 0.14 2.23 0.11 5% 0.16 2.57 0.07 3% 0.18 2.78 0.06 3% 0.12 1.84 0.17 1%

3042210 0.75 0.18 2.84 0.05 5% 0.14 2.24 0.11 5% 0.17 2.63 0.07 5% 0.18 2.81 0.06 4% 0.12 1.83 0.18 0%

3042310 0.5 0.18 2.80 0.06 4% 0.14 2.20 0.11 3% 0.16 2.58 0.07 3% 0.18 2.75 0.06 2% 0.12 1.83 0.18 0%

3042410 0.25 0.17 2.67 0.07 12% 0.14 2.12 0.12 7% 0.15 2.33 0.10 7% 0.15 2.42 0.09 8% 0.11 1.78 0.19 6%

3042101 1 0.16 2.56 0.07 0.12 1.82 0.18 0.15 2.30 0.10 0.15 2.42 0.09 0.10 1.51 0.26

3042201 0.75 0.17 2.63 0.07 0.11 1.76 0.19 0.14 2.26 0.10 0.15 2.43 0.09 0.09 1.37 0.31

3042301 0.5 0.17 2.67 0.07 0.11 1.80 0.18 0.15 2.36 0.09 0.16 2.52 0.08 0.10 1.59 0.24

3042401 0.25 0.13 2.10 0.13 0.09 1.34 0.32 0.10 1.63 0.23 0.11 1.74 0.20 0.08 1.23 0.38

3042111 1 0.16 2.57 0.07 0% 0.12 1.88 0.16 4% 0.15 2.28 0.10 -1% 0.15 2.39 0.09 -1% 0.10 1.58 0.24 4%

3042211 0.75 0.17 2.71 0.06 3% 0.11 1.78 0.19 1% 0.14 2.26 0.10 0% 0.16 2.56 0.07 5% 0.09 1.42 0.29 4%

3042311 0.5 0.18 2.81 0.06 5% 0.12 1.82 0.18 1% 0.15 2.35 0.09 0% 0.16 2.50 0.08 -1% 0.09 1.49 0.27 -7%

3042411 0.25 0.15 2.38 0.09 14% 0.10 1.51 0.26 13% 0.12 1.92 0.16 18% 0.13 2.05 0.13 18% 0.08 1.29 0.35 5%

3042102 1 0.16 2.53 0.08 0.11 1.76 0.19 0.14 2.27 0.10 0.15 2.34 0.10 0.09 1.49 0.27

3042202 0.75 0.16 2.50 0.08 0.10 1.58 0.24 0.14 2.20 0.11 0.14 2.24 0.11 0.09 1.38 0.31

3042302 0.5 0.16 2.54 0.08 0.10 1.54 0.25 0.14 2.18 0.11 0.15 2.29 0.10 0.08 1.23 0.38

3042402 0.25 0.13 1.98 0.15 0.07 1.09 0.45 0.10 1.50 0.27 0.10 1.61 0.23 0.06 0.97 0.52

3042112 1 0.16 2.58 0.07 2% 0.11 1.77 0.19 0% 0.15 2.28 0.10 0% 0.15 2.32 0.10 -1% 0.10 1.55 0.25 4%

3042212 0.75 0.17 2.68 0.06 7% 0.11 1.67 0.21 6% 0.15 2.36 0.09 7% 0.15 2.43 0.09 8% 0.09 1.40 0.30 2%

3042312 0.5 0.17 2.64 0.07 4% 0.10 1.51 0.26 -2% 0.14 2.23 0.11 2% 0.15 2.33 0.10 2% 0.08 1.20 0.39 -3%

3042412 0.25 0.15 2.35 0.09 18% 0.07 1.11 0.44 2% 0.10 1.62 0.23 8% 0.12 1.86 0.17 15% 0.07 1.03 0.48 6%

4 200 psf

No

**0

Yes

No

5%

Yes

No

10%N

on

sim

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Co

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Occ

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Yes

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no

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No

nsi

mu

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d C

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Occ

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Occ

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41.86% Draft


Me

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3102100 1 0.07 2.18 0.11 0.06 1.98 0.15 0.07 2.17 0.12 0.05 1.48 0.27

3102200 0.75 0.07 2.17 0.12 0.06 1.93 0.16 0.07 2.17 0.12 0.05 1.48 0.27

3102300 0.5 0.07 2.17 0.12 0.06 1.94 0.15 0.07 2.17 0.12 0.05 1.48 0.27

3102400 0.25 0.04 1.38 0.31 0.04 1.15 0.42 0.04 1.22 0.38 0.03 1.02 0.49

3102110 1 0.07 2.23 0.11 2% 0.06 1.91 0.16 -3% 0.07 2.18 0.11 1% 0.05 1.42 0.29 -3%

3102210 0.75 0.07 2.22 0.11 2% 0.06 1.90 0.16 -2% 0.07 2.17 0.12 0% 0.04 1.38 0.31 -6%

3102310 0.5 0.07 2.29 0.10 6% 0.06 1.94 0.15 0% 0.07 2.20 0.11 2% 0.04 1.38 0.31 -7%

3102410 0.25 0.05 1.47 0.28 6% 0.04 1.26 0.36 10% 0.04 1.35 0.32 11% 0.04 1.11 0.43 9%

3102101 1 0.07 2.02 0.14 0.05 1.46 0.28 0.06 1.91 0.16 0.04 1.11 0.43

3102201 0.75 0.07 2.21 0.11 0.05 1.52 0.26 0.07 2.01 0.14 0.03 0.96 0.53

3102301 0.5 0.07 2.20 0.11 0.04 1.35 0.32 0.07 2.02 0.14 0.03 1.00 0.50

3102401 0.25 0.04 1.17 0.41 0.03 0.93 0.54 0.03 1.03 0.48 0.03 0.83 0.61

3102111 1 0.07 2.09 0.13 3% 0.05 1.46 0.28 0% 0.06 1.95 0.15 2% 0.04 1.10 0.44 -1%

3102211 0.75 0.07 2.27 0.10 3% 0.05 1.45 0.29 -5% 0.06 1.96 0.15 -3% 0.03 1.02 0.49 6%

3102311 0.5 0.07 2.21 0.11 1% 0.04 1.31 0.34 -3% 0.07 2.04 0.14 1% 0.03 0.93 0.55 -7%

3102411 0.25 0.04 1.33 0.33 14% 0.03 0.94 0.54 1% 0.04 1.15 0.42 11% 0.03 0.86 0.59 4%

3102102 1 0.07 2.04 0.14 0.05 1.40 0.30 0.06 1.89 0.16 0.03 1.06 0.47

3102202 0.75 0.07 2.24 0.11 0.04 1.38 0.31 0.06 2.00 0.14 0.03 1.01 0.49

3102302 0.5 0.07 2.18 0.12 0.03 0.97 0.52 0.06 1.75 0.19 0.03 0.82 0.62

3102402 0.25 0.04 1.15 0.41 0.02 0.72 0.69 0.03 0.99 0.50 0.02 0.70 0.71

3102112 1 0.07 2.05 0.13 1% 0.04 1.38 0.31 -1% 0.06 1.91 0.16 1% 0.04 1.08 0.45 2%

3102212 0.75 0.07 2.24 0.11 0% 0.05 1.40 0.30 1% 0.06 1.97 0.15 -2% 0.03 1.02 0.49 1%

3102312 0.5 0.07 2.28 0.10 5% 0.03 0.94 0.54 -3% 0.06 1.85 0.17 6% 0.03 0.84 0.60 2%

3102412 0.25 0.04 1.24 0.37 8% 0.02 0.68 0.72 -6% 0.03 1.04 0.48 5% 0.02 0.72 0.69 3%

10 200 psf

No

nsi

mu

late

d C

oll

ap

se

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

3%

Yes

No

no

nsi

mu

late

d

Co

lla

pse

Mo

de

s

No

nsi

mu

late

d C

oll

ap

se

Occ

urs

if

IDR

at

Bu

ild

ing

Ed

ge

Ex

cee

ds

6%

No

nsi

mu

late

d C

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ap

se

Occ

urs

if

IDR

at

Bu

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Ed

ge

Ex

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ds

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C

(LV

CC

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com

pu

ted

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mn

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ial

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No

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Yes

No

Na

tura

l E

cce

ntr

icit

y (

Off

set

of

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)

5%

Yes

No

10%

ID

Sto

rie

s

Gra

vit

y

Re

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ve

Fra

me

Sp

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ng

(S

/L)

De

sig

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or

Acc

ide

nta

l T

ors

ion



50% Draft


Table ETable ETable ETable E----15151515 Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1----Story, Low Gravity, Symmetric Story, Low Gravity, Symmetric Story, Low Gravity, Symmetric Story, Low Gravity, Symmetric

Archetypes for which Accidental Torsion was not Archetypes for which Accidental Torsion was not Archetypes for which Accidental Torsion was not Archetypes for which Accidental Torsion was not

Considered in Design (Torsional Modes in Grey)Considered in Design (Torsional Modes in Grey)Considered in Design (Torsional Modes in Grey)Considered in Design (Torsional Modes in Grey)

Mode

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

1 1.40 lat 1.40 lat 1.62 tors 3.98 tors

2 1.40 lat 1.40 lat 1.40 lat 1.40 lat

3 0.78 tors 1.05 tors 1.40 lat 1.40 lat

Table ETable ETable ETable E----16161616 Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1----Story, High Gravity, Symmetric Story, High Gravity, Symmetric Story, High Gravity, Symmetric Story, High Gravity, Symmetric



Mod

e

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

1 1.52 lat 1.52 lat 1.76 tors 4.49 tors

2 1.52 lat 1.52 lat 1.52 lat 1.52 lat

3 0.84 tors 1.13 tors 1.52 lat 1.52 lat

Table ETable ETable ETable E----17171717 Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1----Story, High Gravity, Story, High Gravity, Story, High Gravity, Story, High Gravity, ‘I‘I‘I‘I----Shaped’Shaped’Shaped’Shaped’



Mode

S/L = 0.5 S/L = 0.45 S/L = 0.4

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

1 2.00 tors 2.26 tors 2.62 tors

2 1.52 lat 1.52 lat 1.52 lat

3 1.52 lat 1.52 lat 1.52 lat

Mode S/L = 0.35 S/L = 0.3 S/L = 0.25

1 3.14 tors 3.97 tors 5.65 tors

2 1.52 lat 1.52 lat 1.52 lat

3 1.52 lat 1.52 lat 1.52 lat

41.86% Draft


Table ETable ETable ETable E----18181818 Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1Modal Periods of the 1----Story, High Gravity Archetypes Story, High Gravity Archetypes Story, High Gravity Archetypes Story, High Gravity Archetypes

with Inherent Torsion for which Accidental Torsion was with Inherent Torsion for which Accidental Torsion was with Inherent Torsion for which Accidental Torsion was with Inherent Torsion for which Accidental Torsion was

not Considered in Design (Torsional Modes in Grey)not Considered in Design (Torsional Modes in Grey)not Considered in Design (Torsional Modes in Grey)not Considered in Design (Torsional Modes in Grey)

Mode

S/L = 0.5 S/L = 0.25

Period

(s)

Torsional

or

Lateral

Period

(s)

Torsional

or

Lateral

1 2.28 tors 5.94 tors

2 1.42 lat 1.43 lat

3 1.22 lat 1.33 lat

Table ETable ETable ETable E----19191919 Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4----Story, Low Gravity Symmetric Story, Low Gravity Symmetric Story, Low Gravity Symmetric Story, Low Gravity Symmetric


Considered in Design Considered in Design Considered in Design Considered in Design (Torsional Modes in Grey)(Torsional Modes in Grey)(Torsional Modes in Grey)(Torsional Modes in Grey)

Mod

e

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

1 2.34 lat 2.34 lat 2.70 tors 6.99 tors

2 2.34 lat 2.34 lat 2.34 lat 2.34 lat

3 1.29 tors 1.74 tors 2.34 lat 2.34 lat

4 0.80 lat 0.80 lat 0.93 tors 2.14 tors

5 0.80 lat 0.80 lat 0.80 lat 1.25 tors

6 0.49 lat 0.61 tors 0.80 lat 0.89 tors

7 0.49 lat 0.49 lat 0.56 tors 0.80 lat

Table ETable ETable ETable E----20202020 Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4Modal Periods of the 4----Story, High Gravity Symmetric Story, High Gravity Symmetric Story, High Gravity Symmetric Story, High Gravity Symmetric



Mod

e

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

1 2.34 lat 2.34 lat 2.70 tors 6.78 tors

2 2.34 lat 2.34 lat 2.34 lat 2.34 lat

3 1.30 tors 1.75 tors 2.34 lat 2.34 lat

4 0.83 lat 0.83 lat 0.96 tors 2.21 tors

5 0.83 lat 0.83 lat 0.83 lat 1.32 tors

6 0.51 lat 0.63 tors 0.83 lat 0.91 tors

7 0.51 lat 0.51 lat 0.59 tors 0.83 lat

50% Draft


Table ETable ETable ETable E----21212121 Modal Periods of the 10Modal Periods of the 10Modal Periods of the 10Modal Periods of the 10----Story, High Gravity Symmetric Story, High Gravity Symmetric Story, High Gravity Symmetric Story, High Gravity Symmetric



Mod

e

S/L = 1 S/L = 0.75 S/L = 0.5 S/L = 0.25

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

Perio

d (s)

Torsion

al or

Lateral

1 4.28 lat 4.28 lat 4.95 tors 14.6

8 tors

2 4.28 lat 4.28 lat 4.28 lat 5.24 tors

3 2.32 tors 3.14 tors 4.28 lat 4.28 lat

4 1.67 lat 1.67 lat 1.93 tors 4.28 lat

5 1.67 lat 1.67 lat 1.67 lat 2.90 tors

6 0.98 lat 1.23 tors 1.67 lat 1.89 tors

7 0.98 lat 0.98 lat 1.13 tors 1.67 lat

41.86% Draft


References

ACI 318-10, Building code requirements for reinforced concrete, Detroit:

American Concrete Institute; 2010.

Altoontash, A., 2004, Simulation and Damage Models for Performance

Assessment of Reinforced Concrete Beam-Column Joints, Ph.D.

Dissertation, Department of Civil and Environmental Engineering,

Stanford University, Stanford CA.

Anagnostopoulos1, S.A., Alexopoulou1, C., Stathopoulos, K.G., 2010, “An

answer to an important controversy and the need for caution when

using simple models to predict inelastic earthquake response of

buildings with torsion,” Earthquake Engineering & Structural

Dynamics, 39 (5), pp. 521–540.

Aslani, H., 2005, Probabilistic earthquake loss estimation and loss

disaggregation in buildings, Ph.D. Dissertation, Department of Civil

and Environmental Engineering, Stanford University, Stanford, CA.

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