NIST Technical Note 1940

Database-Assisted Design and 

second-order effects on the windinduced structural behavior of 

high-rise steel buildings

Sejun Park

DongHun Yeo

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NIST Technical Note 1940

Database-Assisted Design and 

second-order effects on the windinduced structural behavior of 

high-rise steel buildings

Sejun Park 

DongHun Yeo 

Engineering Laboratory 

National Institute of Standards and Technology

Gaithersburg, MD 20899-7311 

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October 2016

Includes updates as of 03-07-2017; See Appendix 3

U.S. Department of Commerce 

Penny Pritzker, Secretary

National Institute of Standards and Technology 

Willie May, Under Secretary of Commerce for Standards and Technology and Director 
Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement by the

National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose. 

National Institute of Standards and Technology Technical Note 1940

Natl. Inst. Stand. Technol. Tech. Note 1940, 37 pages (October 2016) 

CODEN: NTNOEF

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i 

Disclaimers

(1) The policy of the NIST is to use the International System of Units in its technical

communications. In this document however, works of authors outside NIST are cited which 

describe measurements in certain non-SI units. Thus, it is more practical to include the nonSI unit measurements from these references.

(2) Certain trade names or company products or procedures may be mentioned in the

text to specify adequately the experimental procedure or equipment used. In no case does 

such identification imply recommendation or endorsement by the National Institute of 

Standards and Technology, nor does it imply that the products or procedures are the best 

available for the purpose. 

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ii

Abstract

This work presents an investigation into second-order effects on the wind-induced 

structural dynamic behavior of high-rise steel structure, as considered within a Databaseassisted Design (DAD) context. A geometric stiffness method that accounts for secondorder effects and allows the dynamic analysis to be performed without iterations is shown 

to be applicable in conjunction with DAD and is used in a study of the response of a 60-

story building, known as the CAARC building. Datasets of the aerodynamic pressure on 

the CAARC building for suburban exposure are used to calculate overturning moments 

and shear forces at the base, as well as members’ demand-to-capacity indexes (DCIs), interstory drift ratios, and resultant accelerations. Dynamic analyses are performed using five 

reference mean hourly wind speeds at the rooftop for suburban terrain exposure (Uref = 20 

m/s, 40 m/s, 60 m/s, 80 m/s, and 100 m/s). The first three and the last three wind speeds 

are used in analyses for serviceability and strength, respectively. The second-order effects 

decrease natural frequencies of vibration of the building by up to 12 %. If the resonant 

response is not taken into account and for Uref = 80 m/s, second-order effects increase the 

non-directional peak base shears by up to 9 %, the torsional moments by up to 10 %, and 

the overturning moments by up to 15 %. If resonance is accounted for, (i) for Uref = 100 

m/s normal to a building face, the vortex shedding frequency is close to the 2nd and the 3rd

natural frequencies of the building, and those effects are increased by 40 % to 56 %; for 

Uref = 80 m/s and a typical set of 21 structural members, the DCIs for the interaction of 

axial forces and bending moments, Bij

PM, are increased by up to 19 % for columns, 41 % 

for beams, and 31 % for diagonal bracings, while the DCIs for the shear forces, Bij

V are 

increased by up to 67 % for columns, 26 % for beams, and 13 % for diagonal bracings; (iii) 

for Uref = 60 m/s, second-order effects increase the inter-story drift ratios by up to 40 % 

and the resultant accelerations at the top floor by up to 20 %. 

Keywords: CAARC building; Database-Assisted Design (DAD); Demand-to-capacity 

index (DCI); Geometric stiffness approach; High-rise steel structures; Second-order 

effects; Wind effects.

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iii

Acknowledgement

The wind tunnel data developed at the CRIACIV-DIC Boundary Layer Wind Tunnel were 

kindly provided by Dr. Ilaria Venanzi of the University of Perugia. Dr. Emil Simiu served 

as project leader and provided helpful comments, and careful review by Dr. John Gross of 

National Institute of Standards and Technology is acknowledged with thanks. 

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iv

Contents

Disclaimers .......................................................................................................................... i

Abstract ............................................................................................................................... ii

Acknowledgement ............................................................................................................. iii

Contents ............................................................................................................................. iv

List of Figures ..................................................................................................................... v

List of Tables ..................................................................................................................... vi

1. Introduction ................................................................................................................. 1

2. The use of the DAD procedure for the evaluation of second-order effects ................. 2

2.1. Geometric stiffness approach for second-order elastic analysis .......................... 2

2.2. Overview of the DAD procedure ......................................................................... 4

3. Case study .................................................................................................................... 5

3.1. Dynamic properties and modal contribution ........................................................ 7

3.2. Dynamic responses considering second-order effects ......................................... 8

3.3. Strength design: Demand-to-Capacity Index (DCI)........................................... 15

3.4. Serviceability design: Inter-story drift and acceleration .................................... 19

4. Conclusions ............................................................................................................... 24

References ......................................................................................................................... 25

Appendix 1. Demand-to-Capacity Indexes for Steel Structures ....................................... 27

Appendix 2. Global responses: Inter-story drift and Acceleration ................................... 28

Appendix 3. Change log ................................................................................................... 29

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v 

List of Figures

Figure 1. Degrees of freedom of 2D beam-column element .............................................. 3

Figure 2. Overview of DAD procedure .............................................................................. 5

Figure 3. Schematic views of structural system and selected members ............................. 6

Figure 4. First six mode shapes........................................................................................... 7

Figure 5. Accumulated modal contributions to resultant top-floor accelerations ............... 8

Figure 6. Aerodynamic load and its dynamic internal responses ....................................... 9

Figure 7. Frequency distributions of wind excitation and overturning moments with 

respect to hourly mean winds (left: along-wind resp.; right: across-wind resp.) .............. 10

Figure 8. Peak effective overturning moment coefficients under along- and across-wind 

responses ........................................................................................................................... 11

Figure 9. Force coefficients as a function of wind direction (Uref = 60 m/s) .................... 13

Figure 10. Moment coefficients as a function of wind direction (Uref = 60 m/s) ............. 14

Figure 11. Response databases: Bij

PM and BijV (member label = CC1) ............................ 15

Figure 12. 2nd order effect ratios for Bij

PM depending on member types and wind speeds18

Figure 13. 2nd order effect ratios for Bij

V depending on member types and wind speeds . 18

Figure 14. Inter-story drifts along the column line (Uref = 40 m/s, θ = 90°) ..................... 20

Figure 15. Peak inter-story drifts with respect to wind directions (Uref = 40 m/s) ........... 21

Figure 16. Resultant accelerations for the column line (Uref = 40 m/s, θ = 90°) .............. 22

Figure 17. Peak top-floor accelerations with respect to wind directions (Uref = 40 m/s) . 23

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vi 

List of Tables

Table 1. Dynamic properties of building with and without second-order effects .............. 8

Table 2. Second-order effects on the Peak effective overturning moment coefficients 

under along- and across-wind responses........................................................................... 12

Table 3. Second-order effects on the peak base shears and overturning moments ........... 14

Table 4. Peak DCI (Bij

PM) for two load combinations and second-order effects .............. 16

Table 5. Peak DCI (Bij

V) for two load combinations and second-order effects ................ 17

Table 6. Second-order effects on the peak inter-story drift ratios and top-floor 

accelerations ...................................................................................................................... 23

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1 

1. Introduction

In high-rise buildings gravity loads cause an amplification of the structural system’s 

displacements and moments induced by lateral loads. The amplification is a second-order 

effect that includes both the P-Δ effect due to member chord rotation and the P-δ effect due 

to member curvature. High-rise buildings, which typically have fundamental frequencies 

of vibration lower than 1 Hz, tend to be more susceptible to second-order effects associated 

with wind than with seismic loads, since wind loads are generally characterized by low 

frequencies while seismic loads usually have higher frequency content. Therefore, 

controlling second-order effects of high-rise buildings subjected to the wind loading is 

necessary from a perspective of both strength design and serviceability. Two basic 

approaches to assessing dynamic instability induced by secondary effects on multi-story 

buildings have been developed to date: (1) second-order elastic analysis with geometric 

nonlinearities and (2) second-order inelastic analysis with geometric and material 

nonlinearities.

Within the framework of the second-order elastic analysis, several methods have been 

proposed. The additional story shear force method, which deals only with the P-Δ effect 

approximately, implements additional story shears due to the vertical loads with regard to 

the deformed geometry of the structure, and subsequent reanalysis should be performed 

iteratively (Wood et al. 1976). The moment amplification method (also known as B1/B2

approach introduced in ANSI/AISC 2010a and LeMessurier 1976, 1977), which involves 

the calculation of B1 (for P-δ) and B2 (for P-Δ), is simple and fast, and is for this reason 

commonly used in design practice. However, the approximations it entails may be 

unsatisfactory in some cases. The fictitious column method proposed by Rutenberg (1981)

considers the secondary effect by using a fictitious member having negative lateral stiffness 

properties proportional to the story weights. This fictitious member reduces the lateral 

stiffness of the structure so that the drifts and moments of members can be functions of the 

lateral loads and of the gravity loads. The method requires no iterations and only slightly 

underestimates the secondary effects even when the P-δ effect is not included. As 

computational capabilities have advanced, matrix analysis approaches have been 

developed. These approaches can accurately account for both P-Δ and P-δ effects by 

employing stability functions (Goto and Chen 1987) or geometric stiffness formulations

(Wilson and Habibullah 1987). If stability functions are used, the governing differential 

equations of a beam-column element are solved iteratively by updating the stiffness matrix 

and the force vector due to the secondary effects (Al-Mashary and Chen 1990). In the 

geometric stiffness formulation, an assumed cubic polynomial shape function is employed 

to solve the governing equations; this is computationally more advantageous than the use 

of stability functions. 

Among second-order inelastic analyses, the pushover analysis have been widely used. 

For simplicity, the inelastic material behavior is typically assumed to be bilinear with zero 

post-yield stiffness (i.e., elastic-perfectly plastic). Several nonlinear dynamic analysis 

techniques for multi- or equivalent single-degree-of-freedom systems are then applied to 

estimate the inelastic dynamic responses. Bernal (1998), MacRae (1993), Tremblay et al. 

(2001), Gupta and Krawinkler (2000), and Humar et al. (2006) carried out inelastic 

dynamic analyses on various models of multi-story buildings designed for seismic loads 

and proposed several methods of accounting for the secondary effects. They reported that 

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2 

the second-order effects typically result in an increase of the response over the first-order 

response by approximately 10 % to 25 %, depending upon the lateral forces resisting 

system, the number of stories, and the magnitude and duration of ground motions. 

Second-order effects on tall buildings have been extensively investigated for the case 

of seismic loads (Gupta and Krawinkler 2000; Humar et al. 2006; MacRae 1993; 

Williamson 2003). However, research for the case of wind loads has been much more 

limited. The ASCE task committee on drift control of steel building structures has 

suggested such research (ASCE 1988). Analytical studies have, therefore, been performed 

on steel frames subjected to wind to assess second-order effects on lateral drift of structures

as they affect serviceability (Baji et al. 2012; Berding 2006). However, these studies did 

not include second-order effects on structural strength. 

The main objective of this report is to study the second-order effects on the windinduced strength and serviceability behavior of a high-rise steel structure. This study

adopted the geometric stiffness approach and, as shown in Section 2, used this approach in 

conjunction with the Database-Assisted Design (DAD) technique to account for secondary 

effects on dynamic structural responses under wind with various speeds and directions. The 

structural system was assumed to behave linearly (i.e., material nonlinearity is not 

considered). In Section 3 that approach was applied to a 60-story building, known as the 

CAARC building model, in suburban exposure, for which the wind load was based on 

aerodynamic pressure datasets obtained in wind tunnel tests. First- and second-order 

responses were evaluated for overturning moments and base shear forces, members’ 

demand-to-capacity indexes (DCIs), inter-story drift ratios, and accelerations. A section on 

conclusions ends this work. 

2. The use of the DAD procedure for the evaluation of second-order 

effects

2.1. Geometric stiffness approach for second-order elastic analysis

In second-order elastic analyses of both P-Δ (member chord rotation effects) and P-δ 

(member curvature effects), static equilibrium is formulated on the deformed configuration 

of the structure. The secondary effects can be accounted for by using a matrix known as 

the geometric stiffness matrix [also called initial stress stiffness matrix, Wilson and 

Habibullah (1987)]. For the frame analysis, the geometric stiffness matrix represents the 

stiffening and weakening effect by the tensile (positive) and compressive (negative) load 

in the structural member, respectively. The method does not require iterations. The secondorder problem can be formulated and solved as a linear system where the geometric 

stiffness matrix is subtracted from the elastic stiffness matrix, as expressed in Eqs. 1 to 3. 

The following example is an application to a two-dimensional beam element with six 

degrees of freedom (Chen and Lui 1987): 

KKF G ][ ∆⋅−= (1)

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3 









































−

−

−

−

⋅=

4

. /6/12

00/

2/604

/6/120/6/12

00/00/

][

2

2 2

sym LL

IA

L

LL LL

IA IA

L

EI K (2)









































−

−−

−

⋅=

152

10156

000

1010152 30

10156010156

000000

][

/L

sym. /L/

/L /L/

/L/ /L/

G PK (3)

Figure 1. Degrees of freedom of 2D beam-column element

where F is the applied static lateral force matrix; K is the elastic stiffness matrix; Δ is the 

element nodal displacement vector; KG is the geometric stiffness matrix, which accounts 

for second-order moments caused by the interaction of the gravity loads (P) and the lateral 

deflections of structure; L is the element length; E is the modulus of elasticity; A and I are 

the cross-sectional area and the inertia moment of the member, respectively (Figure 1). 

In the geometric stiffness method, (1) if the initial axial forces in the elements are 

significantly modified by the application of external loads, iterative calculations may be 

required, and (2) if P/Pe exceeds 0.4, where Pe = π2

EI/L2, the corresponding members must 

be subdivided into two or more elements to limit the errors in the stiffness matrix; this is 

associated only with the P-δ amplification (White and Hajjar 1991). This solution is as 

accurate as the exact solution obtained by a matrix approach based on stability functions

(Al-Mashary and Chen 1990). 

The matrix equation of dynamic equilibrium in a structural system is: 

 G =∆−+∆+∆ )t(F)t(]KK[)t(C)t(M (4)

y

Δ x 1

Δ4

Δ2

Δ3

Δ5

Δ6

L

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4 

where M, C, and F(t) are the mass, damping, and external excitation matrix at time t, 

respectively. Since the lateral stiffness of the structural system is effectively reduced by 

KG, the natural frequencies of vibration will be lowered and the modal shapes will be 

slightly changed in comparison with those of the analysis in which secondary effects are 

not accounted for. These lower frequencies and the corresponding modal shapes represent 

the actual free vibration responses of the structure (Newmark and Rosenblueth 1971). 

Based on the dynamic properties resulting from Eq. 4, the DAD procedure can be used to 

evaluate the structural response to wind and perform the design procedure for a high-rise 

building by considering the second-order effects under wind excitation. 

2.2. Overview of the DAD procedure

Figure 2 illustrates the DAD procedure for the case in which second-order effects on a 

high-rise building are considered. The procedure within the dotted lines box represents the 

main algorithm of the High-Rise Database-Assisted Design (HR_DAD_2.0) software. The 

software HR_DAD_2.0 is similar to the software HR_DAD_RC developed by Yeo (2010), 

with an additional function that accounts for second-order effects. The natural frequencies 

of vibration and mode shapes can then be obtained by a modal analysis that employs a 

finite-element analysis program, and are input into the main DAD algorithm described in 

Fig. 2. In the modal analysis the factored dead and live loads should be used as vertical 

loads (White and Hajjar 1991). A separate second order analysis is required for each 

factored load combination (as shown in Eqs. 5 and 6). In the DAD module, dynamic 

analyses are performed for the building model with a lumped mass, gravity load and a wind 

load on each floor. The wind loads applied at the floors’ mass centers are calculated from 

aerodynamic pressures on the building for five wind speeds and a sufficiently large set of 

wind directions. The outputs of the dynamic analyses consist of (1) sums of time-series of 

the aerodynamic and inertial forces applied at the center of mass in the direction of each of 

the two principal axes of the building, as well as of the sum of the aerodynamic moment 

and inertial moment about the center of mass. (2) time-series of displacements and 

accelerations at each floor level. These results are converted, for example, through the use 

of influence coefficients, into time-series of internal forces, and time-series of demand-tocapacity indexes (DCIs) for each structural member of interest, as well time-series of interstory drifts in the principal directions of the structure along column lines, and resultant 

accelerations at corners of the top floor.

It is recalled that the DCI of a structural member is the left-hand side of the design 

interaction equation. Details on the DCIs employed in this study are provided in Appendix 

1. For design purposes, the peak of the time-series of each DCI is used and can be 

efficiently calculated using the multiple points-in-time (MPIT) approach (Yeo 2013). This 

study uses 30 peaks of individual wind effects (Yeo and Simiu 2011). Directional wind 

effects are calculated by using directional wind speed climatological databases (hurricane 

and/or non-hurricane datasets). The peak wind effects with a specified Mean Recurrence 

Interval (MRI) of, e.g., 700 years or 1700 years, are estimated using a non-parametric 

statistical method (for details see Section 12.4 in Simiu 2011). 

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5 

Figure 2. Overview of DAD procedure 

3. Case study

Figure 3 depicts the 60-story high-rise steel structure being considered, with 182.88 m

height, 45.72 m width, and 30.48 m depth, known as the CAARC (Commonwealth 

Advisory Aeronautical Research Council) building, studied by various researchers 

(Melbourne 1980; Simiu et al. 2008; Venanzi 2005). Wind direction is defined by the 

clockwise angle θ, with the positive x-axis parallel to the long dimension, and the y-axis

parallel to the short dimension of the building cross section. The building has an outrigger 

system to resist the lateral load similar to the structural system studied by Simiu et al. 

(2008) and consists of 2,100 columns, 3,480 beams, and 2,560 diagonal bracings. Columns 

and beams are classified into three types as corner, external, and core for columns, and 

external, internal, and core for beams, respectively. Diagonal bracings are divided into two 

types as core and outrigger bracings. Each type of structural member (column, beam, 

bracing member) has the same dimensions for ten successive floors of the building’s 60 

floors. The columns and bracings consist of built-up hollow structural sections (HSS), and 

the beams consist of rolled W-sections selected from the Steel Construction Manual of 

Preliminary design

Modeling of structure

□ Structural properties

□ Dynamic properties

Performing dynamic analysis

□ Effective lateral loads

□ Displacements

□ Accelerations

Building response database

□ Overturning moment

□ Demand-to-capacity index

□ Inter-story drift

□ Acceleration

Analysis of directional responses

Determination of Peak wind effects

with specified MRIs

Appropriate 

design?

Modal analysis 

considering second-order 

effects

Aerodynamic pressure

database

Influence coefficients

□ Internal forces induced by 

unit load

Climatological database

Micro-meteorological data

End

Redesign

HR_DAD_2.0

yes

no

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6 

AISC (ANSI/AISC 2010b). The yield strength of steel for all members is 250 MPa. The 

modal damping ratios were assumed to be 1.5 % in all modes considered in this study. 

Figure 3. Schematic views of structural system and selected members

The building was assumed to have suburban terrain exposure. Time series of 

aerodynamic loads on each floor were calculated from the pressure data, with wind 

directions in 10° increments measured in wind tunnel experiments at the Prato (Italy) InterUniversity Research Centre on Building Aerodynamics and Wind Engineering (CRIAC 

IV-DIC) Boundary Layer Wind Tunnel (Venanzi 2005). 

Under the assumption of linear elastic structural behavior, dynamic analyses of the 

building were carried out for serviceability and strength. Load combinations associated 

with gravity and wind loads with MRIs specified by ASCE 7-10 (ASCE 2010). Two load 

combination cases (LC1 and LC2 in Eq. 5) were employed for strength design (Section 2.3 

in ASCE 2010) and one (LC3 in Eq. 6) was employed for serviceability design (see 

Commentary Appendix C in ASCE 2010): 

++ 0.10.12.1 WLD (LC1) and + 0.19.0 WD (LC2) (5)

++ 0.15.00.1 WLD (LC3) (6)

where D is the dead loads, L is the live loads, and W is the wind loads, respectively. 

(a) 3D view (b) Front view (c) Side view

45.72 m 30.48 m

x

y

rz

Corner columns

CC1, CC3, CC5

External columns

CE1, CE3, CE5

Core columns

CO1, CO3, CO5

External beams

BE1, BE3, BE5

Internal beams

BI1, BI3, BI5

Column line

(along the building height)

(d) Plan view and selected members and column line

(θ = wind direction)

θ

Core beams

BO1, BO3, BO5

Outrigger and Belt truss system

(located on 20th, 21st, 40th, 41st, and 60th story)

Building core

Wind

Core bracings

XO1, XO3, XO5

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7 

3.1. Dynamic properties and modal contribution 

Once initial dimensions of members in the building were obtained, the modal analysis was 

conducted with and without the second-order effects to calculate the respective natural 

frequencies and mode shapes using SAP 2000 v.17 (Computers and Structures Inc. 2015). 

Figure 4 and Table 1 show the mode shapes and the corresponding natural frequencies up 

to the 6th mode, respectively. The first mode corresponds to drift along the y-axis, the 

second to drift along the x-axis, the third to rotation along the z-axis. The following fourth, 

fifth, and sixth modes correspond to the second mode of the y-dir. translational motion, the 

rotational motion, and the x-dir. translational motion, respectively. As shown in Table 1, 

the second-order effects decrease the natural frequencies of the building by up to 12 %. As 

a result, the frequencies of the 2nd and the 3rd modes become 0.164 Hz and 0.165 Hz, which 

differ by less than 1%.

For the investigation of modal contributions to structural responses to wind, dynamic 

analyses by the HR_DAD_2.0 algorithm based on the modal superposition method were 

performed with 12 accumulated mode cases from the 1st mode up to the 12th mode (i.e., 1st, 

1st – 2nd, ... 1st – 12th). For example, for the resultant top-floor acceleration at a building 

corner, Figure 5 represents the ratios of the peak resultant accelerations with and without 

second-order effects in the accumulated mode cases to their counterparts without the 

effects in the modes up to 12th mode. The accelerations were calculated at a corner of the 

top-floor under mean hourly wind speed at the top floor level of 60 m/s with θ = 90°. As 

shown in the figure, the acceleration ratios without second-order effects are 0.74 for the 

first two modes, 0.93, 0.96, 0.99, and 1.00 for the first three, four, five, and six modes. A 

similar trend is shown in the analysis with second-order effects, except for an 

approximately 7 % increase of their magnitudes in comparison with the analysis without 

second-order effects. These results imply that it is reasonable to use the first six modes in 

the modal superposition analysis for accurately assessing the dynamic responses of the 

high-rise building model to wind considered in this study, with or without accounting for 

second-order effects.

Figure 4. First six mode shapes 

(a) 1st translational 

mode in y-dir.

3rd mode 4th mode 5th mode 6th mode

x y x y x y x y x y x y

(b) 1st translational 

mode in x-dir.

2 1 nd mode st mode

(c) 1st rotational 

mode in z-dir.

(d) 2nd translational 

mode in y-dir.

(e) 2nd rotational 

mode in z-dir.

(f) 2nd translational 

mode in x-dir.

z z z z z z

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Table 1. Dynamic properties of building with and without second-order effects 

Mode 1st 2nd 3rd 4th 5th 6th

Natural freq. (Hz)

1st order (A) 0.165 0.174 0.188 0.503 0.505 0.516 

2nd order (B) 0.154 0.164 0.165 0.478 0.484 0.498 

Ratio (B/A) 0.93 0.94 0.88 0.95 0.96 0.97 

Figure 5. Accumulated modal contributions to resultant top-floor accelerations

3.2. Dynamic responses considering second-order effects 

Dynamic analyses under LC1 (Eq. 5) were performed using five reference wind speeds 

(Uref = 20 m/s, 40 m/s, 60 m/s, 80 m/s, and 100 m/s; mean hourly wind speeds at the rooftop 

of the building with suburban terrain exposure). Although the probability of attaining a 100 

m/s wind speed is not considered in current building codes, this speed is included in this 

study for illustrative purposes. These wind speeds can be converted to 3-sec gust wind 

speeds at 10 m elevation over open terrain exposure resulting in approximately 22 m/s, 43 

m/s, 65 m/s, 87 m/s, and 108 m/s, respectively. The first three wind speeds were selected 

for serviceability design analysis since they correspond to typical ASCE-based basic wind 

speeds with MRI up to 100 years (see Figures CC-1 to CC-4 in ASCE 7-10 2010), and the 

last three wind speeds were chosen for strength design analysis because they are as high as 

the basic wind speeds for Occupancy Category III and IV buildings with MRI up to 1,700 

years (see Figure 26.5-1B in ASCE 7-10 2010). Note that the basic wind speed for 

Occupancy Category IV buildings near Miami, Florida is approximately 90 m/s 

corresponding to MRI = 3,000 years in the ASCE 7-16 draft (ASCE 2016). Though the 

wind directions considered in this study are θ = 0° to θ = 350° in increments of 10°, for 

reasons of symmetry only directions from θ = 0° to θ = 90° need to be used. 

In the analysis of dynamic response to wind using Eq. 4, the effective force applied to 

the mass center of the nth floor of the structure can be calculated as:

tCtMtPtF )()()()( n nn nn

eff

n ∆−∆−=  (7)

Contributed modes

1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 1-11 1-12

Modal contribution ratio

0.4

0.6

0.8

1

1.2

1.4

Top-floor acc.

1 st order

2 nd order

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9 

where the terms in the right-hand side are the forces related to wind (Pn), inertia (Mn), and 

damping (Cn) matrices at each nth floor, respectively. Note that the damping force is 

negligibly small in comparison with the other forces. Figure 6 shows time histories of the 

wind aerodynamic loads [P(t)] and the effective dynamic responses (base shear) in the 

analyses with and without second-order effects when the wind of Uref = 80 m/s approaches 

the building with wind direction of θ = 90° (i.e., the direction parallel to the short dimension 

of the building). In the plots the loads are normalized based on the absolute peak of the 

loads obtained in the analysis with no second-order effects. The horizontal solid lines 

represent positive or negative peak loads with second-order effects. It is seen that the 

across-wind response with the second-order effects is larger than that without the effects 

while the second-order effect is not significant for the along-wind response. 

Figure 6. Aerodynamic load and its dynamic internal responses 

The second-order effects on the overturning moment in the along-wind and the acrosswind directions were investigated. The effective overturning moment and the windinduced overturning moments were calculated by summing up moments at the base 

induced by the effective force and by the wind force alone, acting at all floors. Figure 7 

shows the spectral densities of the along-wind and across-wind overturning moments at 

wind speeds Uref = 20 m/s, 60 m/s, and 100 m/s with wind direction θ = 90°. The solid blue 

(dim gray shown in grayscale) and orange (gray) areas represent the effective overturning 

moments in the analyses without and with second-order effects, respectively, and the solid 

gray (light gray) area indicate the overturning moments induced by the wind forces alone. 

In the case of along-wind overturning moments (left plots in Figure 7), noticeable 

background responses are significant. The effective overturning moments show resonant 

responses at the natural frequencies at which the directional wind generates dominant 

vibration modes while the wind-induced moment does not have a resonant part. Note that 

the resonance response in the effective moment occurs at a lower frequency in the secondorder analysis than that in the first-order analysis. In the case of across-wind overturning 

moments (right plots in Figure 7), the peak responses of the effective overturning moments 

occur not only at the natural frequencies, but also at a frequency related to vortex shedding. 

The effect of vortex shedding on the across-wind loads is clearly seen under all cases. 

When the reference wind speed is 20 m/s, the peak frequencies of the effective moments 

(i.e., the natural frequencies) are not close to the frequency of the across-wind moment. As 

wind speed increases, however, the vortex shedding frequency gets closer to the natural 

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10 

frequency of the building. Resonance then occurs in the response with second-order effects 

at Uref = 100 m/s (Figure 7c), when the vortex shedding frequency is close to the 2nd and 

the 3rd natural frequencies, both of which are approximately 0.165 Hz. Note that these two 

frequencies are almost identical in the second-order analysis (see Table 1). This explains 

why the across-wind overturning moment in the y direction can increase significantly when 

the 100 m/s wind is acting on the building along the y direction. 

(a) 20 m/s

(b) 60 m/s 

(c) 100 m/s 

Figure 7. Frequency distributions of wind excitation and overturning moments with 

respect to hourly mean winds (left: along-wind resp.; right: across-wind resp.) 

Figure 8 and Table 2 show the second-order effects on the overturning moment 

coefficient. The along- and across-wind peak overturning moments are shown as functions 

of wind speed V and wind direction θ: 

Frequency [Hz]

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M x ) [(N.m) 2 /s -1

]

10 4

0

1

2

3

4

5

6

7

8

9

Structural response (1 st order)

Structural response (2 nd order)

Wind excitation (Background)

Frequency [Hz]

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M y ) [(N.m) 2 /s -1

]

10 4

0

1

2

3

4

5

6

7

8

9

 

 

 

Frequency [Hz]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M x ) [(N.m) 2 /s -1

]

10 6

0

0.5

1

1.5

2

Frequency [Hz]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M y ) [(N.m) 2 /s -1

]

10 6

0

0.5

1

1.5

2

 

Frequency [Hz]

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M x ) [(N.m) 2 /s -1

]

10 6

0

1

2

3

4

5

6

7

8

Frequency [Hz]

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSD (Ovtn M y ) [(N.m) 2 /s -1

]

10 7

0

0.5

1

1.5

2

2.5

3

3.5

4

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11 

, ,

2 2

( , ) or ( , ) ( , ) or ( , ) 1

2

ovtn x ovtn y

Mx My

MV MV

CV CV

V BH

θ θ

θ θ

ρ

= (8)

where Movtn,x and Movtn,y are the overturning moments in x- and y-direction, respectively, ρ

is the air density, B is the wide dimension of the building, and H is the height of the 

building. When the wind direction is 0°, the across-wind moments are larger than the alongwind moments, while both moments at 90° wind direction are nearly equal to a wind speed 

of 80 m/s. In the case of θ = 90° and Uref = 100 m/s, the ratio of the across-wind moment 

with second-order effect to its counterpart without it is 1.56. This is due to the resonance 

induced by vortex shedding when the second-order effects are considered. The case of 90 

m/s wind speed is also included in Figure 8. 

 (a) Wind direction of 0° 

(b) Wind direction of 90° 

Figure 8. Peak effective overturning moment coefficients under along- and across-wind 

responses 

Wind speed [m/s]

20 40 60 80 90 100

C

Mx

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Along-wind resp. (1 st order)

Along-wind resp. (2 nd order)

Across-wind resp. (1 st order)

Across-wind resp. (2 nd order)

Wind speed [m/s]

20 40 60 80 90 100

C

My

0.5

1

1.5

2

2.5

3

3.5

4

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12 

Table 2. Second-order effects on the Peak effective overturning moment coefficients 

under along- and across-wind responses 

θ Along-wind Movtn [N.m] under LC1 Across-wind Movtn [N.m] under LC1 

20 m/s 60 m/s 100 m/s 20 m/s 60 m/s 100 m/s 

0° 1.02 a 0.49 b

1.02 

0.62 

1.01 

0.76 

0.97 

0.52 

1.11 

1.50 

1.06 

1.77 

0.48 c 0.61 0.75 0.54 1.35 1.67 

90° 0.99 

0.90 

0.98 

1.29 

1.07 

1.33 

1.03 

0.52 

1.04 

1.06 

1.56 

3.68 

0.91 1.32 1.25 0.50 1.02 2.36 a

Ratio of peak effective Movtn coefficients with 2nd order effect to the counterpart with 1st order effect (b/c). b

Peak effective Movtn coefficient with 2nd order effect. c

Peak effective Movtn coefficient with 1st order effect.

The second-order effects are investigated for the shear force coefficients (CFx and CFy) 

and the overturning moment coefficients (CMx and CMy) in the x- and y-directions, 

respectively, and torsional moment coefficient (CT) at the base of the structure. The shear 

force and torsional moment coefficients are defined as:

2

( , ) or ( , ) ( , ) or ( , ) 1

2

x y

Fx Fy

FV FV

CV CV

V BH

θ θ

θ θ

ρ

= and 

2 2

(,) (,) 1

2

z

T

T V C V

V BH

θ θ

ρ

= (9)

where Fx and Fy are the base shear forces in the x- and y-direction, respectively, and Tz is 

the base torsional moment in the z-direction. Figures 9 and 10 show those shear force and 

torsional moment coefficients at the base as a function of wind direction in the wind speed 

of Uref = 60 m/s. The symbols represent the mean values of force and moment coefficients 

and the bars crossing these symbols indicate a range from their minimum to maximum 

peak values. Note that the directions for the along-wind and for the across-wind responses 

are 0° and 90° for CFx and CMy, and 90° and 0° for CFy and CMx, respectively. The results 

show no second-order effects on the mean values, however differences in peak values are 

significant and are of interest from a structural design viewpoint. The across-wind shear 

forces and overturning moment fluctuations are stronger than their along-wind 

counterparts. This is due to the vortex-induced wind forces in the across-wind direction. 

The fluctuations with second-order effects are larger in most cases, especially when the 

wind directions are aligned with the principal axes of the building (i.e., θ = 0° and 90°). In 

addition, the torsional moments have the largest negative and positive mean values for θ = 

10° and 70°, respectively. This was also observed by Matsumoto et al. (1998), and was 

attributed to a separation point shift. Wind-induced torsional responses will be significant 

when a building has the mass center at each floor offset from its elastic center.

Table 3 summarizes the effects of secondary action on non-directional peak base shears 

and overturning moments based on the reference wind speeds under LC1. The peak values 

in the table are defined as the non-directional peak values (i.e., the largest of all directional 

peak values calculated from all wind directions). From a practical design viewpoint, it is 

reasonable to use the non-directional peak values for assessing the second-order effects. 

As shown in the table, the second-order effects are generally on the order of 10 % to 15 % 

of the first-order effects at wind speeds up to Uref = 80 m/s. In the case of Uref = 100 m/s, 

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13 

the peak overturning moment in y-direction (Movtn,y) is increased by approximately 50 % 

by the second-order effects when the wind direction is θ = 90°. This is due to the vortexinduced resonance, as seen in Figure 7(c).

(a) CFx

(b) CFy

(c) CT 

Figure 9. Force coefficients as a function of wind direction (Uref = 60 m/s) 

Wind direction [deg.]

0 10 20 30 40 50 60 70 80 90

C

Fx

-3

-2

-1

0

1

2

3

1 st order

2 nd order

 

3

 

 

 

 

Wind direction [deg.]

0 10 20 30 40 50 60 70 80 90

C

Fy

-3

-2

-1

0

1

2

3

 

 

 

Wind direction [deg.]

0 10 20 30 40 50 60 70 80 90

C T

-0.2

-0.1

0

0.1

0.2

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14 

(a) CMx

(b) CMy

Figure 10. Moment coefficients as a function of wind direction (Uref = 60 m/s) 

Table 3. Second-order effects on the peak base shears and overturning moments

Forces 

and 

moments

Reference wind speeds (Uref) under LC1

20 m/s 40 m/s 60 m/s 80 m/s 100 m/s 

Fx 0.98a 1.94E+06b

0.91

1.32E+07

1.08

3.16E+07

1.09

6.76E+07

1.52

2.86E+08

1.98E+06c 1.45E+07 2.91E+07 6.20E+07 1.88E+08

Fy 1.02

3.55E+06

0.92

1.55E+07

1.03

4.12E+07

1.06

8.54E+07

1.04

1.35E+08

3.48E+06 1.69E+07 4.02E+07 8.08E+07 1.29E+08

Tz 0.92

1.56E+07

1.11

8.80E+07

0.99

2.23E+08

1.10

4.76E+08

1.40

1.23E+09

1.70E+07 7.90E+07 2.25E+08 4.35E+08 8.74E+08

Movtn,x 1.00

3.43E+08

0.89

1.57E+09

1.11

5.07E+09

1.15

1.08E+10

1.06

1.66E+10

3.43E+08 1.77E+09 4.56E+09 9.33E+09 1.56E+10

Movtn,y 1.03

1.95E+08

0.91

1.47E+09

1.04

3.57E+09

1.09

7.66E+09

1.56

3.45E+10

1.89E+08 1.61E+09 3.44E+09 7.03E+09 2.21E+10

a Ratio of peak non-directional base shear or moment with 2nd order effect to the counterpart with 1st order effect(= b/c). b Peak base shears or moments with 2nd order effect. c Peak base shears or moments with 1st order effect.

Wind direction [deg.]

0 10 20 30 40 50 60 70 80 90

C

Mx

-2

-1

0

1

2

1 st order

2 nd order

 

 

Wind direction [deg.]

0 10 20 30 40 50 60 70 80 90

C

My

-2

-1

0

1

2

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15 

3.3. Strength design: Demand-to-Capacity Index (DCI) 

Response databases of demand-to-capacity indexes were calculated in the two load 

combination cases (LC1 and LC2 in Eq. 5) for 21 selected structural members: 9 columns, 

9 beams, and 3 diagonal bracings (see Figure 3d) consisting of 1) three core columns (CO1, 

CO3, CO5), three corner columns (CC1, CC3, CC5), and three external columns (CE1, 

CE3, CE5), on 1st, 21st, and 41st stories, 2) three external beams (BE1, BE3, BE5), three 

internal beams (BI1, BI3, BI5), and three core beams (BO1, BO3, BO5), on 10th, 30th, and 

50th floors, and three core bracings (XO1, XO3, XO5) on 1st, 21st, and 41st stories. Their 

DCIs for interaction of axial forces and bending moments (Bij

PM) and for shear forces (BijV) 

were calculated with wind directions (θ = 0°, 10°, ..., 350°) and wind speeds (Uref = 60 m/s, 

80 m/s, and 100 m/s). Figure 11 shows an example of the response databases of Bij

PM and 

Bij

V for the corner column at the 1st story (CC1) under LC1. The DCI values in the response 

databases are the peak values of a time-series of DCIs calculated in Eqs. A1 to A4. 

Tables 4 and 5 summarize the second-order effects on non-directional peak DCIs for 

the selected members in the two load combination cases, and Figure 12 and 13 illustrate 

their second-order effect ratios. Overall, the maximum Bij

PM values are larger in LC1 than 

in LC2, but the maximum Bij

V values in LC1 are as high as those in LC2. In Table 4, most 

of the peak Bij

PM values are over unity, and they are even higher than 9 for a certain wind 

speed. This is due not only to use of consistent structural members of the building under 

various wind speeds, but also to the limitations of the elastic analysis performed in this 

study. 

(a) Bij

PM from 1st order analysis (b) BijPM from 2nd order analysis

(c) Bij

V from 1st order analysis (d) Bij

V from 2nd order analysis

Figure 11. Response databases: Bij

PM and BijV (member label = CC1)

Wind speeds [m/s]

100

80

60

0

Wind directions [

90 °

]

180

270

360

4

3

2

1

B PM

ij (1

st order)

Wind speeds [m/s]

100

80

60

0

Wind directions [

90

°

]

180

270

360

1

2

3

4

5

B PM

ij (2

nd order)

Wind speeds [m/s]

100

80

60

0

Wind directions [

90 °

]

180

270

360

0.2

0.1

0

0.3

B V

ij (1

st order)

Wind speeds [m/s]

100

80

60

0

Wind directions [

90 °

]

180

270

360

0.2

0.3

0

0.1

B V

ij (2

nd order)

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16 

Table 4. Peak DCI (Bij

PM) for two load combinations and second-order effects

Member Label Story

(floor)

Reference wind speed (Uref) 

60 m/s 80 m/s 100 m/s

Ratio a LC1 (LC2) Ratio LC1 (LC2) Ratio LC1 (LC2)

Corner

column

(CC)

CC1 1st 1.02 1.04 (0.85) b

1.19 1.84 (1.48) 1.66 4.49 (4.29)

1.02 (0.83) c 1.54 (1.35) 2.71 (2.51)

CC3 21st 1.04 0.99 (0.81) 1.14 1.66 (1.32) 1.61 4.18 (4.00)

0.95 (0.77) 1.45 (1.27) 2.60 (2.41)

CC5 41st 1.04 0.92 (0.74) 1.14 1.54 (1.22) 1.65 3.86 (3.65)

0.89 (0.71) 1.35 (1.17) 2.34 (2.16)

Exterior

column

(CE)

CE1 1st 1.04 0.99 (0.76) 1.15 1.54 (1.30) 1.65 4.21 (3.97)

0.96 (0.72) 1.33 (1.10) 2.55 (2.31)

CE3 21st 1.03 0.91 (0.70) 1.17 1.42 (1.21) 1.66 4.12 (3.91)

0.88 (0.67) 1.22 (1.01) 2.49 (2.28)

CE5 41st 1.03 0.92 (0.70) 1.14 1.41 (1.19) 1.62 4.06 (3.84)

0.90 (0.67) 1.23 (1.01) 2.51 (2.29)

Core

column

(CO)

CO1 1st 1.04 2.05 (1.83) 1.16 3.77 (3.54) 1.60 10.17(9.94)

1.98 (1.75) 3.26 (3.03) 6.34 (6.11)

CO3 21st 1.07 1.45 (1.24) 1.15 2.53 (2.32) 1.62 6.80 (6.59)

1.36 (1.15) 2.20 (2.00) 4.18 (3.98)

CO5 41st 1.05 1.23 (0.98) 1.16 2.06 (1.81) 1.62 4.83 (4.57)

1.17 (0.91) 1.77 (1.51) 2.99 (2.73)

Internal

beam

(BI)

BI1 10th 1.12 1.09 (0.81) 1.25 1.65 (1.36) 1.30 2.30 (2.07)

0.98 (0.69) 1.31 (1.03) 1.77 (1.48)

BI3 30th 1.11 1.27 (0.93) 1.23 1.83 (1.49) 1.26 2.52 (2.34)

1.14 (0.80) 1.49 (1.15) 2.01 (1.67)

BI5 50th 1.08 1.23 (0.86) 1.09 1.62 (1.25) 1.10 2.07 (1.93)

1.14 (0.77) 1.49 (1.11) 1.89 (1.51)

External

beam

(BE)

BE1 10th 1.21 0.80 (0.67) 1.41 1.46 (1.33) 1.30 2.26 (2.13)

0.66 (0.53) 1.04 (0.91) 1.74 (1.61)

BE3 30th 1.25 0.87 (0.74) 1.39 1.57 (1.44) 1.18 2.17 (2.04)

0.70 (0.56) 1.13 (1.00) 1.84 (1.71)

BE5 50th 1.20 0.74 (0.60) 1.24 1.23 (1.10) 1.14 1.68 (1.55)

0.62 (0.48) 0.99 (0.86) 1.48 (1.34)

Core

beam

(BO)

BO1 10th 1.09 0.81 (0.52) 1.15 1.06 (0.80) 1.62 2.47 (2.21)

0.74 (0.49) 0.92 (0.67) 1.52 (1.27)

BO3 30th 1.07 0.76 (0.48) 1.11 0.95 (0.69) 1.58 2.09 (1.84)

0.71 (0.45) 0.85 (0.60) 1.33 (1.07)

BO5 50th 1.02 0.68 (0.41) 1.05 0.79 (0.54) 1.39 1.42 (1.16)

0.67 (0.41) 0.76 (0.50) 1.02 (0.76)

Core 

bracing

(XO)

XO1 1st 1.05 0.52 (0.47) 1.08 0.93 (0.89) 1.30 2.16 (2.13)

0.49 (0.46) 0.87 (0.84) 1.67 (1.62)

XO2 21st 1.13 0.64 (0.61) 1.23 1.34 (1.31) 1.22 2.05 (2.02)

0.57 (0.52) 1.10 (1.05) 1.68 (1.64)

XO3 41st 1.16 0.96 (0.92) 1.31 2.01 (1.96) 1.14 2.92 (2.88)

0.83 (0.77) 1.54 (1.47) 2.55 (2.49) a The larger value of ratios of peak non-directional DCIs with 2nd order effect to the counterpart with 1st order effect, in

respective LC1 and LC2.

b Peak DCI with 2nd order effect. c Peak DCI with 1st order effect.

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17 

Table 5. Peak DCI (Bij

V) for two load combinations and second-order effects

Member Label Story

(floor)

Reference wind speed (Uref) 

60 m/s 80 m/s 100 m/s

Ratio a LC1 (LC2) Ratio LC1 (LC2) Ratio LC1 (LC2)

Corner

column

(CC)

CC1 1st 1.07 0.037 (0.036) b

1.30 0.089 (0.088) 1.77 0.258 (0.254)

0.034 (0.034) c 0.068 (0.067) 0.145 (0.146)

CC3 21st 1.48 0.067 (0.065) 1.67 0.148 (0.145) 1.97 0.476 (0.477)

0.045 (0.043) 0.089 (0.086) 0.241 (0.239)

CC5 41st 1.46 0.076 (0.070) 1.48 0.153 (0.147) 1.96 0.458 (0.461)

0.052 (0.046) 0.103 (0.098) 0.234 (0.231)

Exterior

column

(CE)

CE1 1st 1.11 0.038 (0.037) 1.32 0.090 (0.089) 1.50 0.199 (0.186)

0.034 (0.034) 0.068 (0.068) 0.133 (0.133)

CE3 21st 1.18 0.054 (0.051) 1.42 0.121 (0.121) 1.79 0.386 (0.448)

0.046 (0.042) 0.085 (0.082) 0.252 (0.249)

CE5 41st 1.11 0.068 (0.059) 1.11 0.124 (0.118) 1.77 0.443 (0.504)

0.061 (0.052) 0.112 (0.103) 0.290 (0.281)

Core

column

(CO)

CO1 1st 1.07 0.093 (0.092) 1.18 0.207 (0.207) 1.70 0.681 (0.681)

0.087 (0.087) 0.175 (0.175) 0.400 (0.400)

CO3 21st 1.38 0.160 (0.160) 1.39 0.342 (0.342) 1.93 1.273 (1.272)

0.116 (0.116) 0.246 (0.247) 0.660 (0.660)

CO5 41st 1.35 0.093 (0.092) 1.37 0.198 (0.199) 1.89 0.717 (0.714)

0.069 (0.068) 0.146 (0.145) 0.379 (0.377)

Internal

beam

(BI)

BI1 10th 1.05 0.409 (0.266) 1.13 0.507 (0.364) 1.18 0.623 (0.490)

0.388 (0.245) 0.448 (0.305) 0.528 (0.386)

BI3 30th 1.05 0.439 (0.287) 1.13 0.540 (0.388) 1.16 0.663 (0.539)

0.417 (0.265) 0.479 (0.327) 0.572 (0.420)

BI5 50th 1.04 0.434 (0.275) 1.05 0.503 (0.344) 1.06 0.582 (0.466)

0.418 (0.259) 0.479 (0.321) 0.551 (0.392)

External

beam

(BE)

BE1 10th 1.11 0.251 (0.181) 1.26 0.370 (0.300) 1.22 0.511 (0.440)

0.226 (0.157) 0.295 (0.225) 0.419 (0.349)

BE3 30th 1.13 0.264 (0.194) 1.25 0.388 (0.318) 1.13 0.496 (0.426)

0.233 (0.163) 0.310 (0.240) 0.437 (0.368)

BE5 50th 1.10 0.240 (0.170) 1.15 0.328 (0.258) 1.10 0.408 (0.338)

0.219 (0.149) 0.286 (0.216) 0.372 (0.302)

Core

beam

(BO)

BO1 10th 1.03 0.358 (0.220) 1.06 0.403 (0.265) 1.35 0.653 (0.515)

0.346 (0.208) 0.378 (0.240) 0.485 (0.347)

BO3 30th 1.02 0.349 (0.211) 1.04 0.382 (0.244) 1.30 0.586 (0.449)

0.340 (0.202) 0.366 (0.228) 0.450 (0.312)

BO5 50th 1.01 0.335 (0.197) 1.02 0.355 (0.218) 1.18 0.467 (0.329)

0.332 (0.195) 0.349 (0.211) 0.396 (0.258)

Core 

bracing 

(XO)

XO1 1st 1.05 0.032 (0.028) 1.11 0.056 (0.052) 1.80 0.175 (0.172)

0.031 (0.026) 0.050 (0.046) 0.097 (0.093)

XO2 21st 1.05 0.016 (0.012) 1.13 0.028 (0.024) 1.84 0.093 (0.089)

0.016 (0.012) 0.024 (0.020) 0.050 (0.046)

XO3 41st 1.10 0.017 (0.012) 1.11 0.024 (0.019) 1.55 0.052 (0.047)

0.016 (0.011) 0.021 (0.016) 0.033 (0.028) a The larger value of ratios of peak non-directional DCIs with 2nd order effect to the counterpart with 1st order effect, in

respective LC1 and LC2.

b Peak DCI with 2nd order effect. c Peak DCI with 1st order effect.

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18 

Figure 12. 2nd order effect ratios for Bij

PM depending on member types and wind speeds 

Figure 13. 2nd order effect ratios for Bij

V depending on member types and wind speeds 

As shown in Tables 4-5 and Figures 12-13, the second-order effects on axial loads and 

moments (i.e., Bij

PM) of all columns are less than 7 % for Uref = 60 m/s. However, they 

increase by up to 19 % for Uref = 80 m/s and by approximately 66 % for Uref = 100 m/s. In 

the case of Bij

PM for beams, the second-order effects increase by up to 30 % for external 

and internal beams (BE and BI) for all reference wind speeds. However, the second-order 

effects on the core beam (BO) is much larger, up to 62 %, than for the other selected beams.

For Uref = 100 m/s with θ = 90°, the DCI values of all columns, core beams, and core

bracings increase by 14 % to 66 % because vortex-induced across-wind fluctuations lead 

to increased axial forces and bending moments in their members.

Member label

CC CO CE BI BE BO XO CC CO CE BI BE BO XO CC CO CE BI BE BO XO

Ratio for B PM

ij

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2 60 m/s 80 m/s 100 m/s

1 st story (10 th floor)

21 st story (30 th floor)

41 st story (50 th floor)

Member label

CC CO CE BI BE BO XO CC CO CE BI BE BO XO CC CO CE BI BE BO XO

Ratio for B V

ij

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2 60 m/s 80 m/s 100 m/s

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19 

The demand-to-capacity indexes for shears, Bij

V, are less than 0.7 in all cases, 

considerably lower than Bij

PM. The second-order effects on all columns increase with wind 

speeds up to 48 %, 67 %, and 97 % for Uref = 60 m/s, 80 m/s, and 100 m/s, respectively. In 

the case of beams, the second-order effects are up to 35 % larger for all wind speeds. Note 

that the Bij

V values of most members are considerably smaller than the BijPM, which means 

Bij

V is not the critical factor in the structural design of the building considered in this study.

3.4. Serviceability design: Inter-story drift and acceleration 

Response databases for inter-story drifts and accelerations were calculated in the load 

combination case of LC3 (Eq. 6) along a column line of interest for serviceability design 

(see Figure 3d). The reference wind speeds were taken as 20 m/s, 40 m/s, and 60 m/s for 

the analysis for serviceability. Details on expressions for the inter-story drift ratio and the 

acceleration for the building are provided in Appendix 2. Figure 14 shows the inter-story 

drift ratios in both x- and y-direction corresponding to the across- and the along-wind 

response, respectively, along the column line when the reference wind speed is Uref = 40 

m/s and the wind direction is θ = 90°. As shown in the figure, the second-order effect 

increases the inter-story drifts by up to 30 % and 17 % in the along- and across-wind 

response, respectively. Note that the inter-story drift ratios are less than 0.001 on 20th, 21st, 

40th, and 41st stories where the outrigger and belt truss systems are located. Figure 15 plots 

the peak inter-story drift ratios from all stories as a function of wind direction and shows 

that the second-order effect increases inter-story drifts in most wind directions. Since the 

selected column line is located at the lower right-hand corner of the building plane (see 

Fig. 3d), the peak inter-story drifts for x- and y-direction is symmetry with the wind 

directions of 90° and 270° in Fig. 15a and with 180° in Fig. 15b, respectively. Note that 

the x-direction inter-story drift at wind direction of 10° is larger than that of 0°, which can 

be explained by the reattachment of wind flows on the rectangular section of the building 

as previously mentioned in Section 3.2. The behavior is also observed in the y-direction

inter-story drift for wind directions between 80° and 90°. 

Figures 16 and 17 show the resultant accelerations for the column line at the wind 

direction of 90° and the top-floor accelerations with respect to wind directions, 

respectively, when the reference wind speed is 40 m/s. The unit of acceleration used in the 

figures is milli-g, where g is the gravitational acceleration (9.81 m/s2). As shown in the 

Figure 16, the second-order effect can be noticeable only above approximately the top half 

floors and increases top-floor accelerations by 2 %. 

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20 

(a) Along-wind response (b) Across-wind response 

Figure 14. Inter-story drifts along the column line (Uref = 40 m/s, θ = 90°) 

Inter-story drift ratio (y) 10 -3

0 1 2 3 4 5 6

Story

1

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

Along-wind

1 st order effect

2 nd order effect

 

 

 

 

 

 

Inter-story drift ratio (x) 10 -3

0 1 2 3 4 5 6

Story

1

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

Across-wind

1 st order effect

2 nd order effect

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21 

(a) Along the x-direction

 (b) Along the y-direction

Figure 15. Peak inter-story drifts with respect to wind directions (Uref = 40 m/s) 

Wind direction [deg.]

0 45 90 135 180 225 270 315 360

Inter-story drift ratio (X-dir)

10 -3

0.5

1

1.5

2

2.5

3

3.5

4

1 st order

2 nd order

 

 

Wind direction [deg.]

0 45 90 135 180 225 270 315 360

Inter-story drift ratio (Y-dir)

10 -3

0.5

1

1.5

2

2.5

3

3.5

4

1 st order

2 nd order

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22 

Figure 16. Resultant accelerations for the column line (Uref = 40 m/s, θ = 90°) 

Acceleration [mg]

0 10 20 30 40 50

Floor

1

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

Roof

1 st order effect

2 nd order effect

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23 

Figure 17. Peak top-floor accelerations with respect to wind directions (Uref = 40 m/s) 

Table 6 summarizes the peak inter-story drift ratios and resultant accelerations of the 

selected column line (Figure 3d) as a function of reference wind speeds, and their secondorder effects. The peak values in the table are defined in a manner similar to the peak base 

shears and overturning moments (see Table 3), as the largest of all directional peak interstory drift ratios and acceleration values calculated from all wind directions. Based on the 

results in this case study, the second-order effects increase the inter-story drift ratios by 14 

% to 40 % and the resultant accelerations by 2 % to 20 %, respectively. Note that such 

second-order effects are almost constant at Uref = 20 m/s and 40 m/s, but that they are 

substantially increased at the highest wind speeds (Uref = 60 m/s). 

Table 6. Second-order effects on the peak inter-story drift ratios and top-floor accelerations 

Serviceability

factors

Reference wind speed (Uref) under LC3

20 m/s 40 m/s 60 m/s

In-dr. ratio in x 1.21 a 0.00046 b1.14 

0.00336 

1.25 

0.00748 

0.00038 c 0.00293 0.00599 

In-dr. ratio in y 1.17 

0.00077 

1.17 

0.00360 

1.40 

0.01256 

0.00066 0.00309 0.00898 

Resultant acc.d 1.06 

5.37 

1.02 

44.50 

1.20 

162.22 

5.06 43.53 134.87 

a Ratio of peak serviceability factors with 2nd order effect to the counterpart with 1st order effect. b Peak serviceability factors with 2nd order effects. c Peak serviceability factors with 1st order effects. d Unit: milli-g

Wind direction [deg.]

0 45 90 135 180 225 270 315 360

Top-floor acceleration [mg] 10

15

20

25

30

35

40

45

50

1 st order effect

2 nd order effect

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24 

4. Conclusions

This work presents an investigation into second-order effects on the wind-induced 

structural dynamic behavior of high-rise steel structure, as considered within a DatabaseAssisted Design (DAD) context. A geometric stiffness method that accounts for secondorder effects and allows the dynamic analysis to be performed without iterations is shown 

to be applicable in conjunction with DAD and was used in a study of the response of a 60-

story building, known as the CAARC building. Datasets of the aerodynamic pressure on 

the CAARC building for suburban exposure were used to calculate overturning moments 

and shear forces at the base, as well as members’ demand-to-capacity indexes (DCIs), interstory drift ratios, and resultant accelerations. Under the assumption of linear elastic 

structural behavior, dynamic analyses of the building were performed for serviceability and 

strength. The structural behavior was analyzed using global effects (overturning moments, 

base shear forces, and torsion), as well as local effects: i) for strength design, demand-tocapacity indexes (DCIs) of structural members, ii) for serviceability design, inter-story drift 

ratios, and resultant accelerations along a column line. Those values were obtained both by 

considering and disregarding wind directional effects. Of five reference wind speeds at the 

rooftop of the building (Uref = 20 m/s, 40 m/s, 60 m/s, 80 m/s, and 100 m/s), the first three 

were used for serviceability analysis, and the last three for strength analysis. The following 

conclusions can be drawn from this study: 

(1) The second-order effects decrease natural frequencies of vibration of the building 

by up to 12 %. As a result, the 2nd and the 3rd natural frequencies become close to less than 

1 %. The first six modes in the modal superposition analysis were shown to be sufficient 

for accurately assessing the dynamic responses, both when considering and when 

disregarding second-order effects.

(2) When the wind approaches the building in the principal directions of the building, 

the second-order effects increase the along-wind peak effective overturning moments by 

up to 7 % in all wind speeds. However, the across-wind counterparts are increased by up 

to 11 % in wind speeds of 20 m/s and 60 m/s, and by 56 % in the 100 m/s wind speed. The 

latter case is significantly influenced by the vortex-induced resonance phenomenon when 

the vortex shedding frequency is close to the 2nd and the 3rd natural frequencies of the 

building. 

(3) For non-directional second-order effects, the peak base shears are increased by up 

to 9 %, the torsional moments by up to 10 %, and the overturning moments by up to 15 %, 

in the non-resonant cases. However, they are increased by 40 % to 56 % in the resonant 

case of the 100 m/s wind speed and the 90° direction. 

(4) For secondary effects on strength of structural members, the DCIs for axial force 

and bending moments (Bij

PM) are increased by up to 19 % for columns, 41 % for beams, 

and 31 % for diagonal bracings, and those for shear forces (Bij

V) by up to 67 % for columns, 

26 % for beams, and 13 % for diagonal bracings in the case of Uref = 80 m/s. For Uref = 100 

m/s, for which across-wind resonance occurs, the increments due to the second-order

effects are 66 %, 62 %, 30 %, 97 %, 35 %, and 84 % respectively. Note that the Bij

V values 

of most members are considerably smaller than the Bij

PM, which means BijV is not the critical 

factor in the structural design of the building considered in this study. 

5) The second-order effects increase the inter-story drift ratios and the resultant 

accelerations by up to 40 % and 20 %, respectively, in the 60 m/s wind speed. The inter-

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25 

story drift ratios show the secondary effects along all stories except ones where the 

outrigger and belt truss systems are located. However, the secondary effects on the resultant 

acceleration are shown above approximately the top half floors. 

While much research was performed on secondary effects on high-rise building 

subjected to earthquake loads, this work is, to the authors’ knowledge, the first study to 

focus on the systematic analysis of second-order effects on high-rise buildings subjected 

to wind loads. The estimates presented in this study were performed to the CAARC 

building model. However, they show that second-order effects on structural responses of 

high-rise buildings to wind loads should be analyzed at the design stage. Future research is 

recommended on estimates of second-order effects that take into account material 

nonlinearities as well as beam-column joint models in various types of main wind force 

resisting systems. 

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LeMessurier, W. J. (1976). "A practical method of second order analysis, Part 1 - Pinjointed frames." Engineering journal, AISC, 13(4), 89-96. 

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Appendix 1. Demand-to-Capacity Indexes for Steel Structures

The demand-to-capacity index (DCI) is the left-hand side of the design interaction 

equation and is a measure of the degree to which structural members are designed 

adequately for strength (Simiu 2011). The final design for strength is achieved when the 

DCIs of members are as close as possible to unity, to within specified serviceability 

constraints. The available strengths in this report are based on the AISC 360-10 

(ANSI/AISC 2010a), and they depend on the cross-sectional shapes and types of member 

(i.e., column, beam, or bracing). 

A. Members subjected to flexure and axial forces (Bij

PM) 

When ≥ 2.0

n

r

P

P

φ , 0.1

9

8 ≤















+⋅+

ny

ry

nx

rx

n

r

M

M

M

M

P

P

φφφ (A1)

 When < 2.0

n

r

P

P

φ , 0.1

2

≤















++ ⋅ ny

ry

nx

rx

n

r

M

M

M

M

P

P

φφφ (A2)

B. Members subjected to shear (Bij

V) 

≤ 0.1

n

r

V

V

φ (A3)

where Pr is the required axial strength using LRFD load combinations; Pn is the design 

tensile or compressive strength; Mrx is the required flexural strength about the strong axis 

using the LRFD load combinations; Mry is the required flexural strength about the weakaxis using the LRFD load combinations; Mnx is the available flexural strength about the xaxis; Mny is the available flexural strength about the y-axis; Vr is the required shear strength 

using the LRFD load combinations; Vn is the design shear strength; Tr is the required 

torsional strength using the LRFD load combinations; Tn is the design torsional strength; 

φ is the resistance factor for each type of strength.

C. HSS members subjected to combined torsion, shear, flexure, and axial force (Bij

PMVT) 

When the required torsional strength is less than or equal to 20 % of the available 

torsional strength, the interaction of torsion, shear, flexure and/or axial force for HSS 

(Hollowed Structural Section) shall be determined by the Equation (A1) or (A2) and the 

torsional effects shall be neglected. When the required torsional strength exceeds 20% of

the available torsional strength, the interaction of torsion, shear, flexure and/or axial force 

shall be limited, at the point of consideration, by 

01

2

. T

T

V

V

M

M

M

M

P

P

n

r

n

r

ny

ry

nx

rx

n

r ≤ 















++















++

φφφφφ (A4)

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Appendix 2. Global responses: Inter-story drift and Acceleration

The time-series of the inter-story drift ratios at i

th story, di,x(t) and di,y(t), corresponding 

to the x and y axis, are

i

i iyi i iyi

xi h

tDtxtDtx

td )]()([)]()([ )( , 1 1,1

,

− − −− −− = θ θ (A5)

, 1 1, 1

,

[ ( ) ( )] [ ( ) ( )] ( ) i ix i i i x i

i y

i

yt D t y t D t d t

h

+ −+ θ θ − −− = (A6)

where xi(t), yi(t), and θi(t) are the displacements and rotation at the mass center of the i

th

floor, Di,x and Di,y are distances along the x and y axes from the mass center of the i

th floor 

to the point of interest on that floor, and hi is the i

th story height between mass centers of 

the i

th and the (i-1)th floor. 

The time-series of the resultant accelerations at the i

th floor, ai,r(t) is expressed as:

2

,

2

, , tDtytDtxta )]()([)]()([)(ri i θiyi i θixi

    −= ++ (A7)

where accelerations tx )(i

 , ty )(i

 , and t)( θi

 of the mass center at the top floor pertain to 

the x, y, and θ (i.e., rotational) axes, and Di,x and Di,y are the distances along the x and y

axes from the mass center to the point of interest on the i

th floor.

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Appendix 3. Change log

Revision 1 – March 7, 2017 

▪ The influence coefficient matrix for the second-order analysis is calculated from the 

softened stiffness matrix (K−KG). The results influenced by the modified matrix are 

revised in the publication including figures and tables: 

 - Figure 11 (Response databases of Bij

PM and BijV) 

- Figure 12 and 13 (2nd order effect ratios depending on member types and wind speeds) 

- Table 4 and 5 (Peak DCIs for two load combinations and second-order effects)

▪ Revised the peak inter-story drifts with respect to wind directions (Figure 15). 

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