NIST BUILDING SCIENCE SERIES 181

High-Rise Database-Assisted Design 1.1

(HR_DAD_1.1): Concepts, Software, and

Examples

Seymour M.J. Spence
Cover Photo: Boundary Layer Wind Tunnel Laboratory, University of Florence, Prato, Italy.
NIST BUILDING SCIENCE SERIES 181

High-Rise Database-Assisted Design 1.1

(HR_DAD_1.1): Concepts, Software, and

Examples

Seymour M.J. Spence

Building and Fire Research Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8611

February 2009

U.S. Department of Commerce

Dr. Gary Locke, Secretary

National Institute of Standards and Technology

Dr. Patrick D. Gallagher, Deputy Director
Disclaimer

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 Building Science Series 1??

Natl. Inst. Stand. Technol. Bldg. Sci. Ser. 180, 66 pages (December 2008)

CODEN: NBSSES
Abstract

This report documents the changes that have been made to the previous structure of the HighRise Database-Assisted Design (HR_DAD) software, thereby developing the program

HR_DAD_1.1 capable of calculating efficiently the response of tall buildings made up of many

thousands of members. In particular the following upgrades on the previous software have been

implemented: (1) Capability of performing time-domain calculations, requiring times of the

order of hours, of the peak demand in each member of a real size structure (estimated time for

the previous version of HR_ DAD on an AMD Athlon 64 processor 3000+ with 512MB was on

the order of weeks); (2) Include as output of the software the peak demand of global response

parameters such as inter-story drift and top floor acceleration for any number of column lines

and locations on the top floor; (3) Allow time histories of floor displacements and top floor

accelerations to be saved for selected wind directions and speeds; (4) Implement a general input

format for the wind tunnel information concerning the Synchronous Multi-Pressure Sensing

System measurements, therefore allowing for easy consideration of tall buildings with irregular

geometries. (5) Predispose the software for the eventual inclusion of a multi-hazard approach to

wind design of tall buildings, that is, consider various types of wind, including hurricanes,

synoptic winds, and thunderstorms. The techniques, observations and theory that have been used

to achieve these changes are documented together with the validation of the new software. The

report includes a detailed user’s manual.

Key words: Building technology; multi-hazard engineering; structural engineering; tall

buildings; wind engineering.

 iii
Acknowledgments

This work was produced during the author's tenure as a Guest Researcher with the Structures

Group, Building and Fire Research Laboratory (BFRL), on assignment from the Department of

Civil and Environmental Engineering, University of Perugia, Italy. Dr. Massimiliano Gioffrè of

the University of Perugia and CRIACIV (Italian Inter-University Research Center on

Aerodynamics of Construction & Wind Engineering), and Dr. Emil Simiu of BFRL served as

research advisers. The original HR_DAD software for which the enhancements described in this

report were developed was created by Mihai Iancovici of the Bucharest Technical University of

Civil Engineering and BFRL, William P. Fritz, BFRL, and René D. Gabbai, BFRL. Valuable

contributions by Joseph A. Main, BFRL, are acknowledged with thanks. Mircea D. Grigoriu,

Cornell University and BFRL, and Fahim Sadek, BFRL, provided helpful comments and

suggestions on the format and content of the report. Permission to reproduce the cover photo was

kindly given by Claudio Borri, Director, CRIACIV, University of Florence, Italy. The

conceptual framework for the time-domain approach used in this work was developed by Emil

Simiu, BFRL.

 iv
Contents

Abstract.......................................................................................................iii

Acknowledgments........................................................................................iv

Contents...................................................................................................v

List of Figures............................................................................................vi

List of Tables.............................................................................................vii

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

2. General Structure of HR_DAD_1.1......................................................2

3. Analysis Framework.........................................................................4

Wind tunnel input (page 2.m)..................................................................4

Mechanical model (program 1.m and

odal_eom_solve.m)...........................................................................................6

4. Global Response Surfaces: Inter-Story Drift and Top Floor Acceleration

 (Program1.M)..................................................................................7

5. Local Response Surfaces: Formulation of the Strength Requirements............8

Influence coefficients...........................................................................8

Interaction formulas.............................................................................9

Formulation based on time history analysis................................................10

Formulation based on the expected peak (program1m).......................................11

6. Modeling of the Wind Directionality Effects and Calculation of the MRIs

(Program2.MAT)..........................................................................13

7. Validation of the HR_DAD_1.1 Software..............................................14

Introduction.....................................................................................14

2D 2-Story Building...................................................... ..................15

Closed form solution.................................................................15

HR_DAD_1.1 solution...............................................................17

Comparison between closed form and numerical results........................20

Summary.........................................................................................20

8. Concluding Remarks........................................................................21

References...............................................................................................22

v
APPENDIX User’s Manual High-Rise Database-Assisted Design 1.1

 (HR_DAD_1.1) Software.........................................................24

 A1 Introduction..................................................................................24

 A2 How to download and install the software..............................................24

 A3 Basics of using the HR_DAD_1.1 software....................................... ....25

 A4 How to use the Manual.....................................................................25

 A5 Definition of the 3D 2-story building....................................................25

 A6 Input for running R_DAD_1.1..............................................................25

 A7 Output from running R_DAD_1.1.................................... ...................27

 A8 Details on the use of the software.........................................................27

List of Figures

Figure 2.1. HR_DAD_1,1 program structure.......................................................3

Figure 4.1. Position parameters Dkjx and Dkjy for floors j and j-1 and

 points kj and kj-1........................................................................7

Figure 4.2. Position parameters DNpx and DNpy for the top floor..................................7

Figure 5.1. Member i..............................................................................................................9

Figure 7.1. 2D 2-story example......................................................................15

Figure 7.2. Comparison of closed form and HR_DAD_1.1 time histories of u1..............20

Figure 7.3.Comparison of closed form and HR_DAD time histories of u2............. ......20

Figure A1. Simple 2-story 3D building.............................................................26

Figure A2. File naming convention used for the floor loads.....................................32

Figure A3. ith member of the structure.............................................................34

Figure A4. Bij_RS file structure.....................................................................47

Figure A5. Response surface associated with member 1.........................................47

Figure A6. InDr_RS_set_X file structure.........................................................48

Figure A7. 2nd x-direction floor response surface of set 1......................................49

Figure A8. Acc_RS_point_X file structure........................................................50

 vi
Figure A9. disp_set_1_40_30(2,:,1), 2nd floor x-direction displacement time history

for column line 1........................................................................54

List of Tables

Table 7.1. Summary of input for running the example (closed form solution)..............16

Table 7.2. Summary of type 1 HR_DAD input for running the example.....................17

Table 7.3. Summary of type 2 HR_DAD input for running the example......................17

Table 7.4. Influence coefficients for running the example.......................................18

Table 7.5. Comparison of various response components.........................................21

Table A1. SAP2000 Excel spreadsheet output for member 1....................................22

vii
viii
1

1. INTRODUCTION

Modern tall buildings are increasingly characterized by irregular geometries. However, this

newly found architectural freedom has complicated their structural design. Traditionally the

design is carried out by considering a fairly large number of load combinations based on

Equivalent Static Wind Loads (ESWLs) derived from High Frequency Force Balance (HFFB)

wind tunnel testing and associated frequency domain analyses. This approach to the design of

tall buildings against wind storms was developed primarily during the 1970s when the possibility

of using Synchronous Multi-Pressure Sensing System (SMPSS) pressure measurements to

characterize the external wind pressure field did not exist, and fast and efficient Ordinary

Differential Equation (ODE) solvers were not available. This is not the case anymore as SMPSS

measurements using up to 500 pressure taps are commonplace as are extremely fast and efficient

ODE solvers.

 The HFFB approach uses the fact that the expression for the base moments in sway

coincides with the respective generalized forces, provided that the fundamental sway modes

depend linearly on height [1, 2]. That approach has the following shortcomings. First, it

measures only the base moments, and therefore provides no information on the effective

distribution of the wind load over the building’s height. This is an obstacle to calculating the

mean and background responses [3, 4], as well as to implementing the various mode correction

schemes required if the fundamental sway modes do not vary linearly with height or are coupled,

as is the case if the center of mass and the elastic center do not coincide [5, 6, 7, 8, 9, 10, 11, 12,

13, 14]. Therefore, in the case of HFFB measurements, the only path is to guess the load

distribution, which introduces approximations that can be particularly crude in the presence of

aerodynamic interference effects due to neighboring tall buildings. Second, HFFB does not

provide the information needed to account for the effect of higher modes of vibration when

calculating the mean, background, and resonant response as only the first three generalized

forces may be estimated. Third, corrections are always required for the fundamental torsional

mode, which would have to vary uniformly with height if the fundamental generalized moment

in torsion were to coincide with the base torsional moment. These corrections depend on

assumptions concerning the wind load distribution on the exterior surface of the building.

Fourth, because the spectral approach commonly used with the HFFB approach entails total loss

of phase information, difficulties arise when attempting to account for joint effects of responses

in the fundamental modes. These difficulties are addressed by considering multiple load

combinations that have little or no theoretical foundation and therefore introduce additional

approximations, as well as massively increasing the designer’s computational tasks. Fifth,

inherent to all frequency domain methods, is the inability to account for non-Gaussian features of

the response, which can be significant [4,15].

All these shortcomings are overcome by adopting a time-domain approach with associated

Synchronous Multi-Pressure Sensing System (SMPSS) pressure measurements [16, 17, 18].

SMPSS pressure measurements allow the distribution of wind loads over the building to be

estimated. Therefore the mean and background responses can be easily and transparently

estimated, automatically accounting for spatial pressure correlations and aerodynamic

interference effects. Also, because the loads over the building height are known, any number of

modes may be considered in the resonant response. This technique is also independent of

whether the building exhibits uncoupled and linear mode shapes or the coupled modes with

nonlinear shapes, and can be applied to structures with simple or complex geometries. The time
2

domain automatically accounts for the correlation between the modal responses, for phase

relationships among various wind effects, and for any non-Gaussian effects.

Time-domain techniques are now possible owing to the availability of fast ordinary

differential equation solvers. Nevertheless, they can be time-consuming, especially when

considering the numerous constraints that must be satisfied in the design of tall buildings. It is

therefore paramount for the success of any time-domain based analysis package to be very

efficient. This will allow the designer to tap into the advantages that are inherent in this

approach.

 This report documents the changes that have been made to the previous structure of the

HR_DAD software (see www.nist.gov/wind, II A, Wind Effects on Flexible Buildings) with the

aim of making this a program capable of calculating efficiently the response of large scale real

world tall buildings made up of many thousands of members. In particular the following

upgrades on the previous software have been implemented:

§ Capability of performing calculations, requiring times of the order of hours, of the

peak demand in each member of a real size structure (estimated time for the previous

version of HR_ DAD on an AMD Athlon 64 processor 3000+ with 512MB was on

the order of weeks).

§ Include as output of the software the peak demand of global response parameters

such as inter-story drift and top floor acceleration for any number of column lines

and locations on the top floor.

§ Allow time histories of floor displacements and top floor accelerations to be saved

for selected wind directions and speeds.

§ Implement a general input format for the wind tunnel information concerning the

SMPSS measurements, therefore allowing for easy consideration of tall buildings

with irregular geometries.

§ Predispose the software for the eventual inclusion of a multi-hazard approach to

wind design of tall buildings, that is, consider various types of wind, including

hurricanes, synoptic winds, and thunderstorms.

In the following sections the techniques, observations and theory that have been used to achieve

these changes will be documented together with the verification of the updated software, which

can be found on Appendix A. The modified software is named HR_DAD_1.1, and is

incorporated in the site www.nist.gov/wind in Section II A under the heading “updated

HR_DAD software (HR_DAD_1.1).”

2. GENERAL STRUCTURE OF HR_DAD_1.1

The general structure of the software has remained similar to that of the previous version,

HR_DAD, documented in www.nist.gov/wind, Section II A, Wind Effects on Flexible Buildings

[16]. The HR_DAD_1.1 software consists of 34 MATLAB files arranged as shown in Fig. 2.1.

The program is started by typing “HR_DAD” at the MATLAB command prompt. This file

initializes all the variables and opens up 10 pages: 2 introductory pages (Page_Main and

Page_Open) and 8 other pages (Page_One through Page_Eight) that run the program. Each page
3

HR_DAD.m

Program1.m

Program2.m HurrData.m

maxmainest.m

Load_Fl.m

Fl_scaled_th.m

F_Load_Vector.m

Modal_eom_solve.m

stdgaminv.m

stdnormcfd.m

stdnorminv.m

Page_Main

Page_Open

Page_One

Page_Two

Page_three

Page_Four

Page_Five

Page_Six

Page_Seven

Page_Eight

contains the eight menu tabs located along the left-hand side of the graphical user interface.

Selection of one of these tabs brings that page to the top of all the opened pages. Pages 1 to 5

contain areas to input values for the variables and open files pertinent to the program. Purple

icons within each page that contain the symbol “?” provide additional information for the

specific item. For example, the menu labelled “1.Building Info.”, which calls the Page_One.m

routine, contains an input box for the variable “RNmodes”. Clicking on the purple information

icon produces a pop-up window that indicates that the variable name within the MATLAB

program is RNmodes, that its matrix size is 1 x 1 (i.e., it is a constant), and that the variable

Figure 2.1. HR_DAD_1.1 program structure

specifies the number of vibration modes considered in the analysis. The user defines the

necessary variables in pages one through five, which automatically change the values of the

(global) variables within the program. The pages and the variables can be selected in any order.

The default values specified in the HR_DAD_1.1.m file are otherwise used.

The actual calculations are performed in page six by selecting the “Run1” and “Run2” buttons,

which run the files called Program1.m and Program2.m, respectively (refer to Fig. 2.1).

Program1.m uses the relative variables to generate the response surfaces, defined in this report as

surfaces whose ordinates are peak responses and whose abscissas are wind speeds and wind

directions, for specified members, inter-story drift (for any number of column lines input on page

five), and top floor accelerations (for any number of locations input on page five). The methods

used to calculate the peak response are differentiated depending on whether a member response

or global response is being considered. In view of the potentially enormous number of members

composing tall buildings (in the order of many thousands) the local member level responses are

calculated by assuming them to be Gaussian. This allows the peak response to be calculated with

great efficiency directly from the covariance matrix of the forces acting on the centers of mass of

each floor. The calculation technique is fully explained in section 5.4. The global response on the

other hand consists of a considerably smaller number (on the order of hundreds) of demands of

interest, therefore allowing the peaks of the relevant times series to be estimated either as

observed peaks or by the modified statistical Rice method, see page five.

For each wind direction WD and wind speed WS (defined in page three) several functions are

used, some of which call other functions as shown in Fig. 2.1. Comments within the MATLAB

script files describe details for these functions. The routine in Program1.m establishes three 3D

arrays of peak values that have dimensions WS × WD × (number of members or column lines

or locations where top floor acceleration is calculated) and saves them under the file locations
4

and names specified in page six. Each face of an array is a discrete representation of the response

surfaces.

The 3D arrays saved in Program1.m are used in Program2.m. The routine in Program2.m cycles

through each member, inter-story drift and acceleration constraint, loads the saved response

surfaces from Run1 (Program1.m), loads the particular hurricane wind speed database (specified

in page five), and calculates the peak wind effect for each MRI (specified in page three and

distinguished between those pertaining to members and those to be used for the global

responses). The peak wind demands as a function of the MRIs are then saved for each member,

inter-story drift set and top floor acceleration point under the file location and name specified in

page six.

Page seven summarizes key input variables while page eight, “8. Save/Load Data”, allows the

user to save the current set of variables to a specified file location or load a previously saved file.

Finally, the “Exit” button closes all pages and exits the HR_DAD_1.1 program. The variables,

however, still remain in memory. The following Sections will outline the theory behind the

various parts of HR_DAD_1.1. Reference will be made to the relevant input quantities used

during the implementation of the routines and to the eventual output.

3. ANALYSIS FRAMEWORK

3.1 Wind tunnel input (page 2.m)

The wind loads and related information are input on page three (Fig. 2.1). The loads

consist of the time histories of the model scale floor loads acting at the center of mass of each of

the N floors. The loads are arranged in a 3N×1 vector { } () () () ()

T

TTT Ft t t t

x y θ =     FFF where

the loads in the directions x, y and θ are given by the sub-vectors

( ) 1 2 ( ) ( ), ( ),..., ( ) T

x x x Nx F t F tF t F t = , ( ) 1 2 ( ) ( ), ( ),..., ( )

T

x y y Ny F t F tF t F t = and

( ) 1 2 ( ) ( ), ( ),..., ( ) T

x N F t F tF t F t = θθ θ . These loads are estimated through wind tunnel tests on rigid

scale models equipped with a number of simultaneously measured pressure taps that allow for

the characterization of the external wind pressure field through non-dimensional pressure

coefficients. It is assumed throughout that aeroelastic effects are negligible. The transformation

of the external pressure field into concentrated loads acting at the centers of mass may be

performed following procedures deemed adequate by the user. The resulting model scale floor

loads must be scaled before they can be applied to the dynamic system. This is done

automatically in HR_DAD_1.1 by respecting the similitude on the reduced frequency:

m m p p

m p

f D f D

V V

= (1)

where D denotes a characteristic dimension of the structure, f denotes the sampling frequency, V

denotes the mean wind velocity at a consistent height (e.g., roof height), and the subscripts m and

p denote “model” and “prototype,” respectively. Letting λL = Dm/Dp denote the length scale of

the wind tunnel model, the prototype sampling frequency can be expressed as follows by

rearranging Eq. (1):
5

p

pm L

m

V

f f

V

λ

 

=    

(2)

The magnitude of the loads must also be scaled. This is simply done by multiplying the

time series of the model loads by ( )2

/ V V p m . All scaling is performed during the execution of

program1.m. The information necessary for this is input in page three.

3.2 Mechanical model (program1.m and modal_eom_solve.m )

The global behavior of tall buildings can be modeled by an equivalent dynamical system

considering each floor to have three degrees of freedom (i.e. x- and y-displacement, relative to

the ground, of the center of mass, and θ-rotation about a vertical axis through the center of

mass). Under these hypotheses the dynamic equilibrium of an N-story building with mass and

stiffness eccentricities which can vary from floor to floor is given by:

[ ] M Ut C Ut K Ut Ft { } () () () () ++ = [ ]{ } [ ]{ }{ } && & (3)

In which:

[ ] M

 

 

=

 

   

M00

0 M0

0 0J

(4a)

[ ] K

 

 

=  

   

xx xy xș

T

xy yy yș

T T

xș \ș șș

KKK

KKK

KKK

(4b)

[ ] C

 

 

=  

   

xx xy xș

T

xy yy yș

T T

xș \ș șș

CCC

CCC

CCC

(4c)

{ } () () () ()

T

TTT Ut t t t =     X Y Ĭ (4d)

where the displacement response sub-vectors are X(t)=(x1(t),x2(t),«,xN(t))T,

Y(t)=(y1(t),y2(t),«,yN(t))T and Θ(t)=(θ1(t),θ2(t),«,θN(t))T; the mass sub-matrix is M=diag[mi] for

i=1,«,N, in which mi is the lumped mass of floor i, while the sub-matrix of mass moment of

inertias of the floor diaphragms is J=diag[Ji] for i=1,«,N, in which Ji represents the polar

moment of inertia of floor i about a vertical axis through the center of mass. Kxx, Kyy, Kθθ, Kyx,
6

Kxθ, Kyθ and Cxx, Cyy, Cθθ, Cxy, Cxθ, Cyθ are the sub-matrixes of the stiffness and damping of the

building respectively.

These generally coupled equations of motion can be solved through modal analysis. In

this framework the modal frequencies ωk

 and mode shapes { } Φk for k=1,2«3N are the roots

and non-trivial solution of the following equations respectively:

2

det( [ ] [ ]) 0 − + ω M K = , or (5a)

{ } 2

( [ ] [ ]) 0 − +Φ ωk k M K = (5b)

From the structure of Eq. 3 it is obvious that if the centers of mass and stiffness for each floor are

coincident and lie on a vertical axis then the system will be uncoupled and the building will

experience uncoupled vibration modes in three principal directions, two orthogonal translational

directions and a rotational direction [1,2]. Otherwise the building will, in general, experience 3D

coupled modes. Due to the orthogonality of the mode shapes, if damping is considered then the

generally coupled system of Eq. 2 maybe transformed into a set of j=1,2,«,3N uncoupled

equations in the generalized coordinates qj:

{ } { }

2

( )

() 2 () ()

[ ]

j

j j jj j j T

j j

Q t

qt qt qt

M

+ +=

Φ Φ

&& && ξω ω (6)

where ξj, and ωj are the damping ratio and circular frequency of mode j, while Qj(t) is the

generalized force given by:

() () { } { } TQ t Ft j j = Φ (7)

Therefore from the knowledge of the global mass matrix, circular frequencies, damping ratios,

mode shapes and scaled time varying floor loads, the modal equations of motion can be solved. It

is this information input on pages 1, 2 and 4 (Fig. 2.1) that is used to define the modal equations

of motion (Eq. 6) that are directly integrated by the MATLAB function modal_eom_solve.m

during the execution of program1.m. The number of modes to be considered in the solution is

user defined. Because the distribution of the wind loads with height is known, any number of

modes may be included in the analysis.

Following this framework the response displacements and accelerations in the global

reference system can be expressed in terms of the contributions of j=1,2,«,n≤3Ν modes. The

displacements for example are given by:

{ } { } Ut qt () [ ] () = Φ (8)

where [ ] Φ is the matrix consisting of the n vectors { } Φ j for j=1,2,«,n≤3Ν while

{} ( ) 1 2 ( ), ( ),..., ( ) T

n

q qt qt qt = denotes the vector of generalized displacements. Likewise the

accelerations are given by:
7

CM

Floor j Floor j-1

kj

x

y

Dkjy Dkjx

CM

kj-1

x

y

Dkj-1y Dkj-1x θ

θ

CM

Top floor

x

y

DNpy DNpx

θ

{ } Ut qt () [ ] () = Φ { } && && (9)

4. GLOBAL RESPONSE SURFACES: INTER-STORY DRIFT AND TOP FLOOR

ACCELERATION (PROGRAM1.M)

From the global response, Eq. 6, it is possible to construct the peak response surfaces for the

inter-story drift and top floor acceleration. For example, given a certain incident wind direction

WD and mean velocity and the top of the building WS, the peak inter-story drift for the kth set of

points and jth floor and in direction s=x,y is given by:

(, ) (, ) (, ) (, )

(, ) 1 11 [ ( ) ( )] [ ( ) ( )]

( )

WS WD WS WD WS WD WS WD

WS WD j kjs j j kj s j

jks

j

x t tx t t

d t

h

θ θ +Δ − +Δ − −−

= (10)

in which Δkjs = −Dkjy for s=x while for s=y Δkjs = Dkjx for j=1,2,«,N, where Dkjx and Dkjy are

shown in Fig. 4.2, kj and kj-1 are the two points belonging to the set k for which the inter-story

drift is to be controlled (Fig. 2.1) and hj is the height of the j

th story.

Figure 4.1. Position parameters Dkjx and Dkjy for floors j and j-1 and points kj and kj-1.

Figure 4.2. Position parameters DNpx and DNpy for the top floor.
8

The top floor acceleration is estimated in a similar fashion. Given a point p=1,2,«,P belonging

to the top floor the acceleration is given by:

(, ) (, ) (, ) () () () WS WD WS WD WS WD

Ns N Ns N a tx t t = +Δ && && θ (11)

in which ΔNp = −DpNy for s=x while for s=y ΔNs = DpNx for p=1,2,«,P, where DpNy and DpNx are

shown in Fig. 4.2.

From the time series of the displacements and accelerations it is possible to evaluate the peak

response in a number of ways. In particular HR_DAD_1.1 software allows the selection (page

five) between considering the observed peak or fitting a marginal distribution to the record

(subroutine maxminest.m). The estimation of the response surface is achieved by solving the

equations of motion for a number of wind directions (WD) and wind speeds (WS). Each surface

is saved in a 3D array, each face of which represents a response surface for a particular

component. The quantities Dkjx , Dkjy and hj for j=1,2,«,N are input via a 3 x (number of sets x

N) matrix on page five (Fig. 2.1) as are DpNy and DpNx via a 2 x (number of points) matrix.

5. LOCAL RESPONSE SURFACES: FORMULATION OF THE STRENGTH

REQUIREMENTS

5.1 Influence coefficients

The time varying internal forces induced in any member by a given incident wind direction γ =

WD and mean velocity at the top of the building V = WS may be obtained from the following

relationships, which for the sake of brevity are shown only for the axial forces and bending

moments:

1

( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )]

N

V V V V V VV

i jxi jx j j jyi jy j j j i j j j

j

N t n F t mx t n F t my t n F t I t

=

= −+ −+ − ∑ && && && γ γ γ γ γ γγ

θ θ θ (12a)

1

( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )]

N

V V V V V VV

iX jXxi jx j j jXyi jy j j jX i j j j

j

M t m F t mx t m F t my t m F t I t

=

= −+ −+ − ∑ && && && γ γ γ γ γ γγ

θ θ θ (12b)

1

( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )]

N

V V V V V VV

iY jYxi jx j j jYyi jy j j jY i j j j

j

M t m F t mx t m F t my t m F t I t

=

= −+ −+ − ∑ && && && γ γ γ γ γ γγ

θ θ θ (12c)

in which ( ) Vq F t js for s=x,y,θ are the components of jth floor load due to the external wind loads

acting at the center of mass while ( ) mx t j j && , ( ) my t j j && and ( ) j j I t θ

&& are the inertial force

components.

The quantities mjXxi, mjYxi, mjXyi, mjYyi mjXθi, mjYθi, njqi, njxi and njyi are the influence

coefficients necessary for the calculation of the internal forces. In particular mjXxi and mjYxi are the

bending moments MjX and MjY about the local axis X and Y of member i (Fig. 5.1) due to a unit

force acting through the center of mass in the x-direction at floor j, while mjXyi and mjYyi are

similar moments due to a unit force acting through the center of mass in the y-direction at floor j;
9

Y

X

Z

Ni

MiY

MiX

Section 1

Section 2

Section 3

mjXθi and mjYθi are the moments due to a unit torque about the mass center at floor j while njqi is

the axial force due to this unit torque; njxi and njyi are the axial forces due to a unit load in the

directions x and y applied to the center of mass of floor j. Obviously the influence coefficients

depend on the section of the member in which they are calculated. HR_DAD_1.1 considers three

sections for each member in which to calculate the internal forces and thereafter verify the

demand-to-capacity index (i.e., the left-hand side of the interaction equation, see Section 5.2) of

the member. In particular the initial, final and mid sections are chosen, that is, sections 1, 2 and 3

of Fig. 5.1.

Figure 5.1. Member i.

HR_DAD_1.1 does not calculate the influence coefficients. They must be obtained

separately using any commercially available structural analysis program such as SAP 2000. This

does not present any particular difficulty. By definition the influence coefficients are simply the

internal forces in three sections of each member due to a statically applied unit load. The only

possible difficulty could lie in the number of load cases, 3N where N is the number of floors,

which must be solved. However this process may be rendered extremely efficient by writing a

subroutine that interacts with the structural analysis program and automatically prepares the

input file for HR_DAD_1.1. It should be observed that even if this not done, due to the extreme

efficacy of modern structural analysis software the calculation of the influence coefficients will

only take a couple of hours or so. Note that the number of influence coefficients is quite

considerable when storage issues are being considered. To avoid problems the coefficients are

arranged in a 3D array, each face of which contains the influence coefficients for the three

sections associated with a particular member. Information on the exact structure of this array

maybe found by pressing on the “?” icon on page three of the HR_DAD_1.1 Graphical User

Interface (GUI) under the heading dynamic influence coefficients.

5.2 Interaction formulas

 In the design of tall buildings, however, it is not individual internal forces that are of interest but

rather combinations of internal forces governed by interaction equations and load combinations.

In the case of steel structures for example, the interaction formulas given by AISC (2001) for the

design of member i are:
10

() () () () 8

0.2 ( ) 1

9

V V VV

if if iXf iYf V

if

ni ni b niX b niY

Nt Nt M t M t

b t

N N MM

 

≥⇒ = + + ≤  

 

γ γ γγ

γ

φ φ φφ

(13a)

() () () ()

0.2 ( ) 1

2

V VV V

if if iXf iYf V

if

ni ni b niX b niY

Nt Nt M t M t

b t

N NM M

<⇒ = + + ≤

γ γγ γ

γ

φ φφ φ

(13b)

where Nni , MniX and MniY are the nominal axial and flexural strengths of member i, φ and φb

are axial and flexural resistance factors while ( ) V

N t if

γ

, ( ) V M t iXf

γ

 and ( ) V M t iYf

γ

 are the total internal

forces due to specified factored combinations (hence the subscript f), with dynamic wind load

contributions calculated from Eqs. 12a-c. Examples of appropriate load combinations can be

found in the ASCE 7-05 Standard:

1.2D + 1.6L (14a)

1.2D + 1.0L + 1.6W (14b)

0.9D + 1.6W (14c)

in which D is the dead load, L is the live load while W is the wind load.

Obviously equations 13a-b represent two distinct time series. Whether 13a or 13b should

be considered depends on the value of ( ) V

Nt N if ni

γ

φ . To construct the response surface it is

necessary to calculate the peak of ( ) V

if b t γ

 for a sufficient number of wind speeds and directions.

In the following two paragraphs methods for calculating these peaks are presented.

5.3 Formulation based on time history analysis

To better understand the structure of ( ) V

if b t γ

it is convenient to rewrite Eqs. 13a-b, dividing them

into a mean static response and a zero mean fluctuating response. For the sake of brevity only

Eq. 13b with combination 14b is shown. The other cases are equivalent.

( )

0.2

V

if

ni

N t

N

< ⇒

γ

φ

( ) V

if b t =

γ

1.6 1.6 1.6

2

VV V ss s

i if iX iXf iY iYf

ni b niX b niY

NN MM MM

NM M

  ++ +   ++ +

 

γγ γ

φφ φ

1.6 1.6 1.6 () () () 1

2

VVV

i iX iX

ni b niX b niY

Nt Mt Mt

NM M

 

++ + ≤    

γ γγ

φφ φ

(15)

in which s Nif ,

s MiXf and s MiYf are the factored internal forces; V

Ni

γ

,

V MiX

γ

 and V MiY

γ

 are the mean

internal forces due to the external wind loading for V H( ) γ while ( ) V

N t i

γ

, ( ) V M t iX

γ

 and ( ) V M t iX

γ

 are

the corresponding zero mean fluctuating components.
11

From the structure of Eq. 15 it is obvious that in order to evaluate ( ) V

if b t γ

 for any number

of combinations and interaction formulas it is necessary to calculate the time histories of the

fluctuating internal forces just once for each section of interest for member i. Once the time

history of ( ) V

if b t γ

 is available the peak may be calculated. In particular, for non-Gaussian time

histories, the method included in the site www.nist.gov/wind, Sect. III B may be used. However,

due to the large number of elements belonging to the structure of a typical tall building, and

given the need to evaluate the response for a sufficient number of wind speeds and directions, the

calculation of the time histories associated with the fluctuating wind load causes a combinatorial

explosion. To avoid this, data compression techniques can be used. The evaluation of the time

histories can be completely avoided if the response can be modeled as Gaussian – which is the

case for effects associated with global contributions to the response, -- leading to a significant

computational advantage. This approach is the method adopted in HR_DAD_1.1 and is fully

explained in the following section.

5.4 Formulation based on the expected peak (program1.m)

If ( ) V

if b t γ

 is assumed to be Gaussian, its expected peak is given by the well known Rice formula

(Davenport, 1964) as:

[ ( )] V

if

V V

if peak if b

bt b g = + γ γ σ γ (16)

in which V

if b

γ

 is the mean response; g is the peak factor while V

if b

σ γ is the standard deviation of

( ) V

if b t γ

. By the definition of the variance, it follows from Eq. 15 that the variance of ( ) V

if b t γ may be

written as:

( )

2

2

12 3 () () () V

if

VVV

i iX iX b

E kN t kM t kM t   = ++    

γγγ σ γ

(17)

where:

1

1.6

2 ni

k

N

=

φ

(18a)

2

1.6

b niX

k

M

=

φ

(18b)

3

1.6

b niY

k

M

=

φ

(18c)

Therefore, in view of the linearity of the expectation operator E, it is possible to the express the

standard deviation as:
12

22 2

1 2 3 12 13 23 V VV V V V V V V V V V V 222

if ii i i iX iX iY iY iX iY iX iY b NN MM MM NM NM MM

σ γ γγ γ γ γ γ γ γ γ γ γ γ

= +++ + + k C k C k C kkC kkC kkC (19)

where V V

N N i i

C γ γ, V V

M MiX iX

C γ γ, V V

M MiY iY

C γ γ, V V

N Mi iX

C γ γ, V V

N Mi iY

C γ γand V V

M MiX iY

C γ γ are the various covariance

coefficients between the fluctuating components of the internal forces. These coefficients can be

calculated directly from the covariance matrix of the global forces Fx acting at the centers of

mass of each floor and the influence coefficients of the internal forces. For example, the

covariance coefficient V V

N Mi i

C γ γ

 between the time varying axial force and bending moment around

the local X axis in member i may be written as:

() () V V

i i

V V

i iX N M

C E N tM t =     γ γ

γ γ (20)

By substituting Eqs. 12a-b into Eq. 20, the right-hand side of Eq. 20 may be rewritten in the

following form:

1

[ ( ) ( )] [ ( ) ( )] [ ( ) ( )]

N

V V V V VV

jxl jx j j jyl jy j j j l j j j

j

E n F t mx t n F t my t n F t I t

=

    −+ −+ −

  ∑ && && && γ γ γ γ γγ

θ θ θ

1

[ ( ) ( )] [ ( ) ( )] [ ( ) ( )]

N

V V V V VV

jXxi jx j j jXyi jy j j jX i j j j

j

m F t mx t m F t my t m F t I t

=

 

  −+ −+ − 

 

∑ && && && γ γ γ γ γγ

θ θ θ (21)

which may in turn be written as:

( )( )

1 1

() () () ()

N N

V VV V

jxi kXxi jx j j kx k k

j k

n m E F t mx t F t mx t

= =

    − −+       ∑ ∑ && && γ γγ γ

( )( )

1 1

( ) ( ) ( ) ( ) ......

N N

V VV V

jxi kXyi jx j j ky k k

j k

n m E F t mx t F t my t

= =

    − −+       ∑ ∑ && && γ γγ γ (22)

where, for example, ( ) () () () () ( ) V VV V E F t mx t F t mx t jx j j kx k k

  − −   && && γ γγ γ is simply the covariance

coefficient between the total forces acting on the center of mass of floors j and k in direction x.

Therefore there is no need to calculate the individual time histories of the internal forces, thus

avoiding a combinatorial explosion.

As previously observed, whether Eq. 13a or 13b should be considered depends on the value

assumed by ( ) V

Nt N if ni

γ

φ at the instant in which the value of ( ) V

if b t γ

is desired. However under the

formulation presented in this paragraph time is eliminated therefore it is not possible to know the

value the of ( ) V

Nt N if ni

γ

φ that accompanies the expected peak of ( ) V

if b t γ

. In HR_DAD_1.1 this

problem is solved by considering the expected maximum and minimum of ( ) V

Nt N if ni

γ

φ that
13

accompany ( ) V

if b t γ

. More precisely if

 max

( ) 0.2 V

if ni

Expected

  Nt N <

 

γ

φ , then only the expected peak

of 13b is considered, while if

 max

( ) V

if ni

Expected

  Nt N   γ

φ and

 min

( ) 0.2 V

if ni

Expected

  Nt N ≥

  γ

φ or

 max

( ) V

if ni

Expected

  Nt N   γ

φ and

 min

( ) 0.2 V

if ni

Expected

  Nt N ≤

  γ

φ , then only 13a is considered. In all

other cases both the expected peaks of 13a and 13b are calculated, with the larger of the two

being saved as representative of the peak demand.

To implement this formulation the influence coefficients must be input. The number of

coefficients depends on the number of sections for which the peak response is to be calculated

along the member. At present in HR_DAD_1.1 a maximum of three sections, the two extreme

sections and the center section, are considered for every member. Therefore for each member a

total of 6 (number if internal forces) × 3 (number of sections to be considered) × N (number of

floors) have to be considered. To input all these coefficients a 3D array is used, in which each

face contains the necessary influence coefficients for the three sections of a particular section.

This array is input on page three (Fig. 2.1). During the execution of program1.m a subroutine

will run through the list of members, also input on page three, and calculate the various

covariance coefficients and the expected peak. To improve speed a distinction is made between

tension and compression elements, and beams and columns, as the number of sections for which

it is necessary to calculate [ ( )] V

if peak b t γ

 depends of the type of element under consideration. For

each wind direction γ = WD and wind speed V = WS only the maximum [ ( )] V

if peak b t γ

occurring in

all three sections is kept and saved as a point of the response surface associated with the

particular member under consideration. The output obtained at the end of running the subroutine

of program1.m dedicated to the calculation of the response surfaces of the individual members is

again a 3D array, in which each face represents the response of the member within the member

list with index coinciding with the third dimension of the array. The location and name of the

array are input on page six.

6. MODELING OF THE WIND DIRECTIONALITY EFFECTS AND CALCULATION

OF THE MRIS (PROGRAM2.MAT)

For design purposes it is necessary to ascertain the peak demand for the various global and local

responses of interest corresponding to specified MRIs. This can be achieved by following the

procedure described in [17, 18], which makes use of a directional extreme wind database such as

the simulated hurricane wind speed database publicly available at www.nist.gov/wind. Work on

a procedure for developing by simulation similar databases for non-hurricane winds is in

progress, and is scheduled for publication and inclusion in the site www.nist.gov/wind in early

2009.

To explain the method consider having p=1,2«,pmax storms, each of which contains the

maximum hourly wind speed for a certain number of directions γ =1,2,«,36. From the peak

response surfaces it is possible to calculate the peak response of each constraint for every

direction and storm [ ] peak

jks p d γ

. From a design point of view, however, it is the maximum value

that occurs during the storm that is of interest. Therefore only the quantities

max [ ] { } peak

jksp jks p d d = γ γ are considered. This means that for each storm there is a single peak
14

response for every constraint. The set of quantities { } jksp d for p=1,2«,pmax represent an

univariate sample of size pmax. By rank-ordering the set { } jksp d it is possible to estimate the peak

response for a given MRI. If the rate of arrival of storms for the site being considered is ν, then

the m-th largest response of interest will have a MRI given by

m

p

MRI

ν

+

=

max

(23)

Therefore by recording the wind direction and mean wind speed at the top of the building for

every data point in the set { } jksp d the direction and mean wind speed can be obtained for any

given MRI and constraint.

The execution of program2.m will initiate a run through all the storm events contained in the

database. The wind directions WD and wind speeds WS for which the structural response has

been calculated during the execution of program1.m do not necessarily coincide with those

contained in the hurricane database. To overcome this problem two interpolation schemes are

available. Method A interpolates linearly between wind speeds and wind directions. Method B

interpolates linearly between wind speeds, but considers only the larger of the two values

corresponding to the two bounding wind directions. Method B is slightly more conservative.

The output files from program2.m depend on the response under consideration. For the structural

members a matrix is output that contains along each row the demand corresponding to the the

MRIs to be estimated. In the case of the inter-story drift a 3D array is used to store the

information. Each face of this array contains the x and y demands corresponding to the MRI

contained in the vector of MRIs input specifically for the inter-story drift and top floor

acceleration. A separate file is generated for every column line input. The demands

corresponding to the top floor accelerations are saved to a matrix that contains the x and y

acceleration demands as columns, while each row corresponds to a particular MRI contained in

the relevant vector. Again a separate file is saved for each point considered.

7. VERIFICATION OF THE HR_DAD_1.1 SOFTWARE

7.1. Introduction

In order to verify the proper functioning of the HR_DAD_1.1 software a simple example will be

solved by HR_DAD_1.1 and compared to the closed form solution. The example represents a

simple 2D 2-story building, assumed to be linearly elastic and have negligible P-ǻ effects. The

solution of the example can be found in [20, Chapter 12], which contains additional details.
15

m

2m

1⁄2k

k

u2

psinωt u1

C1 Sec 1

C1 Sec 2

C1 Sec 3

C3 Sec 1

C3 Sec 2

C3 Sec 3

z1

z2

1⁄2k

k

C2 Sec 1

C2 Sec 2

C2 Sec 3

C4 Sec 1

C4 Sec 2

C4 Sec 3

z1

z2

7.2. 2D 2-story building

7.2.1. Closed form solution

The building is shown in Fig. 7.1. The beams are considered to be rigid. Various responses will

be calculated considering the time dependent load psinωt applied to the first floor and compared

to those obtained from HR_DAD_1.1. The responses that will be compared are: the time

histories of the floor displacements u1 and u2; the peak observed inter-story drift and second floor

peak acceleration; the standard deviation of the peak bending moment occurring in the columns

C1, C2, C3 and C4 (Fig. 7.1).

Figure 7.1. 2D 2-story example.

The closed form solution for the displacements can be found in [20]:

1 12 1 2 ( ) (2 )sin (2 ) cos [ ]

6

p

ut C C t D D t

k

= + ++ ω ω (24a)

2 12 1 2 ( ) (4 )sin (4 ) cos [ ]

6

p

ut C C t D D t

k

= − +− ω ω (24b)

where:

2

22 2

1( )

[1 ( ) ] (2 )

n

n

n nn

C

ω ω

ωω ζ ωω

−

=

− +

 for n =1, 2 (25a)
16

22 2

2

[1 ( ) ] (2 )

n n

n

n nn

D

ζωω

ωω ζ ωω

−

=

− +

 for n =1, 2 (25b)

where 1 ω = k m2 while 2 ω = 2k m . While nζ is the damping ratio of the nth vibrational

mode.

By definition the time histories of the bending moments accruing in columns C1, C2, C3 and C4

will be given respectively by:

( ) 12 1 1 1 ( ,) 2 () M zt z kut C C= = −    (26a)

( ) 2

34 2 2 1

2

( ,) () ()

2

C C

z

M zt k ut ut

=

−   

= −         (26b)

Due to the nature of the structure and loading, the maximum bending moment will occur in the

initial or terminal sections of the columns. Therefore, by choosing the initial section the standard

deviation will be given by:

( ( )) 1 2

2

1 1 2 () 2 () MC C

E ku t E ku t

=

= − −−     σ (27a)

( ( ) ( ) ( )) 3 4

2

21 21 () () () () MC C

E ku t ut E ku t ut

=

  = − − −− −     σ (27b)

where E is the expectation operator.

The maximum observed inter-story drift will be given by [20]:

[ ] ( )( ) 1 2 2

12 1 2

( )

2 2

6

Max u t p

CC DD

h hk

= +++ (28a)

[ ] ( )( ) 2 1 2 2

12 1 2

() ()

22 22

6

Max u t u t p

CC DD

h hk

−

= − +− (28b)

where h is the story height.

The maximum observed top floor acceleration is given by:

( )( )

2 2

2 2 2

2 21 2 1

( ) 4 4

6

du t p Max C C D D

dt k

  ω

= − +−    

(29)

For the worked example the values for the various parameters are listed in Table 7.1.
17

Table 7.1. Summary of input for running the example (closed form solution)

Mechanical and geometrical parameters

Story height h = 4m

Modal damping n

ζ = 1.5% of critical for n = 1,2

Stiffness k = 400000 N/m

Mass m = 15000 kg

p = 10000 N

ω = 1 rad/s

7.2.2 HR_DAD_1.1 solution

HR_DAD_1.1 is developed for analyzing 3D structures. However it is easy to consider a 2D

structure by simply entering as input the relative information for the plane in which it has been

decided to analyze the response. In this example the xz plane is chosen. The forcing function

psinωt is described considering a sampling frequency of 20 Hz for an observation time of 1600 s

and is loaded on page 2 of the HR_DAD_1.1 GUI as a mat file which also contains the time

histories of the loads applied to the other degrees of freedom, which are equal to zero. The input

necessary to run HR_DAD_1.1 is of two types. The first type is directly entered from the GUI

while the second form of input consists of Matlab mat files, e.g., for the forcing function, also

loaded through the GUI. Tables 7.2 and 7.3 summarize the input necessary for running the

example. The variables are organized by their respective pages within the program. The definition of

the variables can be found within the pages of the software and variables not applicable to this

particular wind effect of interest are omitted. The eigenvectors input by the variable evectors (Table

7.3, page 1) come from [20]. Table 7.4 shows the influence coefficients arranged in their 3D array for

the 4 columns of interest.

Table 7.2. Summary of type 1 HR_DAD input for running the example

Page 1 Page 2 Page 3 Page 4

Nfloors = 2

H_bldg = 8

F_dofs = 3

Modes = 6

Modal periods = [1.7207 1 1 0.8604 1 1]

Modal damping = [1.5 1 1 1.5 1 1]

Vm = 1

freq = 20

ms = 1

Npoints = 32001

Nstart = 2000

WS = 1

WD = 0

DLf = 1

SDLf = 1

LLf = 1

WLf = 1

g = 1
18

Table 7.3. Summary of type 2 HR_DAD input for running the example

Page 1 Page 3 Page 4

0.5 0 0 -1 0 0

1 001 0 0

0 100 1 0

0 100 1 0

0 010 0 1

0 010 0 1

evectors

 

 

−

=

−

 

1 1 1 1.111 1

2 1 1 1.111 1

3 1 1 1.111 1

4 1 1 1.111 1

props

 

 

=

 

1234

_

'C' 'C' 'C' 'C'

mem list  =    

30000

1

1

15000

1

1

mass

 

 

 

 

=  

 

 

 

 

Page 4 Page 5

004

_

004

interstory location  =    

acceleration_location = [ ] 0 0

Page 7

1000000000

2000000000

_

3000000000

4000000000

frames DL

 

 

=

 

1000000000

2000000000

_

3000000000

4000000000

frames SDL

 

 

=

 

1000000000

2000000000

_

3000000000

4000000000

frames LL

 

 

=

 

0 0

0 0

displacement_location  =    
19

Table 7.4. Influence coefficients for running the example

Page 3

Elements 1 and 2 Elements 3 and 4

0000 10

0000 10

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

(:,:,1) (:,:,2)

0000 0 0

0000 0 0

0000 0 0

0000 1 0

0000 1 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

dif dif

  −

 

−

= =

 

0000 0 0

0000 10

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

(:,:,3) (:,:,4)

0000 0 0

0000 0 0

0000 0 0

0000 0 0

0000 1 0

0000 0 0

0000 0 0

0000 0 0

0000 0 0

dif dif

 

 

−

= =

 

HR_DAD_1.1 will calculate the element response in the initial, mid and terminal sections as

shown in Fig. 7.1. The maximum response will then be saved to the output file. In the design of

tall buildings it is not the individual internal forces that are of interest, but rather their

combination defined by certain interaction formulas like Eqs. H1-1a and H1-1b contained in the

American Institute of Steel Construction [21] manual for steel structures. Due to the Gaussian

nature of the local member-based response of tall buildings, the peaks of the interaction formulas

can be calculated using the Rice formula. This hypothesis is obviously not valid for this simple

example. However the purpose of this example is not the accurate calculation of its response, but

rather the validation of the calculations being performed by HR_DAD_1.1. Therefore by

specifying an input value of 1.111 Nm for the nominal flexural strengths of the columns and by

ignoring the axial forces in the various columns, the output of HR_DAD_1.1 should yield the

following parameter:

max 1, 2, 3 i

b M i Sec Sec Sec i M = +     σ for i =1, 2,..., 4 (30)
20

100 105 110 115 120 125 130 135 140

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

t(s)

u

1(m)

Closed form solution HR DAD solution

Therefore by considering the particular geometry and loading of the structure,

(sec1 or sec3) i i b M− for i =1, 2,..., 4 , should coincide with the standard deviation of the bending

moment in member i calculated in section 1 or 3 from Eqs. 27a-b.

Once the relative files and variables are defined and loaded, the solution maybe calculated by

simply invoking program1.m by pressing RUN on page 6. The solution will be saved to specified

file locations and names input on pages 6 and 7 of the GUI.

7.2.3. Comparison between closed form and numerical results

Figures 7.2 and 7.3 show the comparison between the time histories of the displacements u1 and

u2 obtained in closed form from Eqs. 24a-b, with those estimated by running HR_DAD_1.1. As

can be seen the numerical results are in good agreement with the exact solution.

Table 7.5 contains the comparison of various response parameters. Good agreement can be seen

in all cases with any differences due to the numerical integration.

7.3. Summary

The purpose of this exercise was to verify the functioning of the HR_DAD software package by

comparing the closed form output from a simple 2D two story shear frame with those obtainable

from running HR_DAD. It was seen that HR_DAD does indeed accurately estimate the response

components necessary for the design of framed structures.

Figure 7.2. Comparison of closed form and HR_DAD time histories of u1
21

100 105 110 115 120 125 130 135 140

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

t(s)

u2(m)

Closed form solution HR DAD solution

Figure 7.3. Comparison of closed form and HR_DAD time histories of u2

Table 7.5. Comparison of various response components

Response parameter Closed form solution HR_DAD

Max σ M

column 1 7495.8 7494.5

column 2 7495.8 7494.5

column 3 146.4060 145.9744

column 4 146.4060 145.9744

Max inter-story drift

floor 1 0.0033 0.0033

floor 2 0.00013 0.00013

Max second floor acceleration 0.0138 0.0139

8. CONCLUDING REMARKS

This report describes the enhancements that have been implemented in HR_DAD in order to make

this a realistic and efficient package for the dynamic time domain structural analysis of tall buildings

subjected to windstorm events. The inherent advantages of working in the time domain include (1)

capability of directly accounting for nonlinear and/or 3D coupled fundamental modal shapes; (2) the

possibility of considering any number of modes in the response; (3) the capability of directly

estimating the correlation that exits between the modal responses; (4) the possibility to include non-
22

Gaussian effects; (5) the preservation of all phase relations, which avoids the need to consider more

than one single combination of load effects, as opposed to the numerous load effect combinations

required if the frequency domain approach is used. A possible drawback of a time domain dynamic

analysis approach is the long computational time necessary compared to more traditional, frequency

domain approaches. Due to the extremely large number of members the processes of calculating

the peak demand for each cross section for a specified MRI could become prohibitive. However,

this is avoided by considering the local response – the response of individual members - to be

Gaussian. The peak response for each wind speed and direction is then calculated from the

covariance matrix of the time-varying global floor loads, thus avoiding the evaluation of time

series of individual internal forces. The global demands, inter-story drift and top floor

acceleration are only on the order of hundreds, therefore allowing the peak response to be

calculated directly from the time series and, therefore, the inclusion of non-Gaussian effects.

This approach to the evaluation of the peak response has allowed HR_DAD_1.1 to become an

efficient and competitive alternative to traditional methods of analysis.

This report contains the results from the completion of the first objective of a far more

ambitious plan. Indeed, it is the first step toward the definition and resolution of a member size

optimization problem formulated in the time domain for the design of tall buildings subjected to

both local strength requirements and global inter-story drift constraints. Due to the enormous size

of this optimization problem, which entails tens of thousands of non-linear constraints, the use of

generic optimizers is out of the question. The only way to efficiently solve the problem is

through specifically designed algorithms. The most appropriate problem definition and

resolution algorithms have been the subject of research carried out in parallel with the continued

development of HR_DAD_1.1. Details can be found in [19]. At present these algorithms are in

the process of being implemented in the HR_DAD_1.1 environment and should be available in

the next version of the software.

REFERENCES

[1] Kareem A., Dynamic Response of High-Rise Buildings to Stochastic Wind, Journal of Wind

Engineering and Industrial Aerodynamics, 41-44, pp. 1101-1112, 1992.

[2] Chen X., Kareem A., Dynamic Wind Effects on Buildings with 3D Coupled Modes:

Application of High Frequency Force Balance Measurements, ASCE Journal of Engineering

Mechanics. Vol. 131, No. 11, pp.1115–1125, 2005.

[3] Spence S.M.J., Gioffrè M., Gusella V., Higher mode contributions to the response of tall

buildings with regular and irregular profiles, 10° Convegno Nazionale di Ingegneria del

Vento, IN-VENTO-2008, Cefalù (PA), 8-11 Giugno 2008.

[4] Spence S.M.J., Gioffrè M., Gusella V., Influence of higher modes on the dynamic response

of irregular and regular tall buildings, 6th International Colloquium on Bluff Bodies

Aerodynamics and Applications (BBAA VI), Milan, Italy, July 20-24, 2008.

[5] P.J. Vickery, A.C. Steckley, N. Isyumov, B.J Vickery. The effect of mode shape on the wind

induced response of tall buildings, Proceedings of the 5th United States National Conference

on Wind Engineering, Lubbock, Texas, 1B-41-1B-48, 1985.

[6] Boggs, D. W., and Peterka, J. A. Aerodynamic model tests of tall buildings. ASCE Journal

of Engineering Mechanics, Vol.115 N. 3, pp. 618-635, 1989.

[7] Xu, Y. L., and Kwok, K. C. S. Mode shape corrections for wind tunnel tests of tall buildings.

Engineering Structures, Vol. 15, pp. 618-635, 1993.
23

[8] Zhou, Y., Kareem, A., and Gu, M. Mode shape corrections for wind load effects. ASCE

Journal of Engineering Mechanics, Vol. 128 N. 1, pp. 15-23, 2002.

[9] Holmes, J. D., Rofail, A., and Aurelius, L. High frequency base balance methodologies for

tall buildings with torsional and coupled resonant modes. Proc., 11th International Conf. on

Wind Engineering, Texas Tech Univ., Lubbock, Tex., pp. 2381-2388, 2003.

[10]Chen, X., and Kareem, A. Coupled building response analysis using HFFB: Some new

insights. Proc., 5th Bluff Body Aerodynamics and Applications (BBAAV), Ottawa, 2004.

[11]Ohkuma, T., Marukawa, H., Yoshie, K., Niwa, H., Teramoto, T., and Kitamura, H.

Simulation method of simultaneous time-series of multi-local wind forces on tall buildings

by using dynamic balance data. Journal of Wind Engineering and Industrial Aerodynamics,

54-55, pp. 115-123, 1995.

[12]Yip, D. Y. N., and Flay, R. G. J. A new force balance data analysis method for wind

response predictions of tall buildings. Journal of Wind Engineering and Industrial

Aerodynamics, 54-55, 457–471, 1995.

[13]Solari, G., Reinhold, T. A., and Livesey, F. “Investigation of wind actions and effects on the

leaning Tower of Pisa.” Wind & Structures, 1-1, pp.1-23, 1998.

[14]Chen, X., and Kareem, A. Validity wind load distribution based on high frequency force

balance measurements. ASCE Journal of Structural Engineering, 131 no. 6, 984-987, 2005.

[15]Xie J., Progress of Wind Tunnel Techniques in Practical Applications, The 4th International

Conference on Advances in Wind and Structures AWAS’08, Jeju, Korea, 29-31, May, 2008.

[16]Main,J. A. and Fritz, W. A., Database-Assisted Design for Wind: Concepts, Software, and

Examples for Rigid and Flexible Buildings, NIST Building Science Series 180, 2006.

[17]Simiu, E. and Miyata, T., Design of Buildings and Bridges for Wind. Wiley, Hoboken, New

Jersey, 2006.

[18] Simiu, E., Gabbai, R.D. and Fritz, W.P.,Wind-induced tall building response: a time-domain

approach. Wind and Structures, Vol. 11, No. 6, 427-440, 2008.

[19]Spence S.M.J., Gioffrè M., A Database-Assisted Design approach for lateral drift

optimization of tall buildings, 14th IFIP WG 7.5 Working Conference on Reliability and

Optimization of Structural Systems (IFIP 08-WG 7.5), Mexico City, Mexico, August 6-9,

2008.

[20]Chopra A. K. Dynamics of structures: Theory and applications to earthquake engineering.

2nd ed. Prentice-Hall; 2000.

[21]American National Standard ANSI/AISC 360-05, Specification for Structural Steel

Buildings. American Institute of Steel Construction, Inc. Chicago, Illinois; 2005.
24

Appendix

User¶s Manual

High-Rise Database-Assisted Design 1.1 (HR_DAD_1.1)

Software

Ssoftware available at www.nist.gov/wind

Disclaimer: Certain trade names or company products are specified in this document to specify adequately the

procedure used. Such identification does not imply recommendation or endorsement by NIST, nor does it imply that

the product is the best available for the purpose. The “stand-alone” version of this software requires installation of

the MATLAB1 Component Runtime (MCR) Libraries provided by The MathWorks, Inc. The author’s limited rights

to the deployment of this program are limited by a license agreement between NIST and The MathWorks. The

license agreement can be found at www.mathworks.com/license/. The author, NIST, and The MathWorks and its

licensors are excluded from all liability for damages or any obligation to provide remedial actions.

1MATLAB®. © 1984 ± 2005 The Mathworks, Inc.

A1. Introduction

This User’s Manual is designed to assist the user of the software HR_DAD_1.1. The software

calculates the response of tall buildings subjected to wind loads, including internal forces in

members, member interaction formulas based on demand-to-capacity ratios, inter-story drift, and

accelerations, for any specified mean recurrence interval of the wind effect being considered.

A2. How to download and install the software

The HR_DAD_1.1 software has been developed using the MATLAB® language and can

be accessed via the internet site http://www.nist.gov/wind. Within the site, click the link “Wind

Effects on Flexible Buildings”. This opens the main page “HR_DAD_1.1 - DAD Software for

High-Rise Buildings” (direct access:

http://www.itl.nist.gov/div898/winds/hr_dad_1.1/hr_dad_1.1.htm). The files available for

download are all in the bulleted list under the heading “Files Available for Download”. In the

following, reference is made to each set of files to be downloaded by the name of the associated

bullet. From the list, first consider the “Files for HR_DAD_1.1 software” bullet. Next to the title

of the bullet item, there is a link to the self-extracting file zip file “HR_DAD_1.1.exe”, which

contains the stand-alone executable. To run HR_DAD_1.1 the MCRInstaller must first be

installed by executing the application “MRCInstaller.exe”. The HR_DAD_1.1 software can then

be launched by double-clicking the file “HR_DAD.exe” within the “stand-alone” folder. This

action opens the ten figure windows (‘Page_Open’, ‘Page_Main’, ‘Page_One’, ‘Page_Two’ ...

‘Page_Eight’) that form the graphical user interface (GUI).
25

A3. Basics of using the HR_DAD_1.1 software

The ten figure windows (i.e., the “pages”) opened above are used primarily to (1) assign values

to the variables used by the HR_DAD_1.1 software (‘Page_One’ through ‘Page_Five’), (2) to

perform the calculations (‘Page_Six’). Variable values can be assigned in any order in

‘Page_One’ through ‘Page_Five’. The variable names within HR_DAD_1.1 are typically shown

in parentheses before the input box on each page. In several instances, a saved MATLAB mat

file is opened within a page to load variables that contain vectors, matrices, or 3D arrays. Help

icons, , are located next to the input boxes for key variables. For a given variable, clicking on

the associated help icon will open a separate window that provides information such as the

variable name, the required variable size, a description of the variable and the specific

organization of the variable’s contents (for vector, matrix or 3D array variables).

A4. How to use the Manual

This manual is organized with the aim of guiding the user through the necessary steps in

order to run HR_DAD_1.1. This is done by first defining a simple 3D 2-story building. The

input necessary for running this example in HR_DAD_1.1 is then schematically outlined,

allowing the user to get an idea of the software’s operation. Each page of the graphical user

interface (GUI) is then described in detail, together with what the input would be in the case of

the simple 3D 2-story building.

A5. Definition of the 3D 2-story building

The structure is depicted in Fig. A1. The floors are considered rigid and no rotation is allowed

between the floor slabs and the 8 columns (members 1 to 8).

A6. Input for running HR_DAD_1.1

To run HR_DAD_1.1 information is necessary concerning the structure, the wind climate at an

open site at a reasonable distance from the building, and the relation between the directional

speeds at 10 m elevation at that site and the corresponding directional wind speeds (including

veering effects, see [17, p. 15], as follows:

1. Information related to the dynamic properties of the structure:

§ Mass and mass moment of inertia with respect to the center of mass, for each floor..

§ Mode shapes, frequencies, and damping ratios. The mode shapes must be

referenced to the centers of mass of each floor.

2. Information for member demand-to-capacity ratio calculations:

§ Influence coefficients, defined as the internal force that occurs in a given section of

a given member due to a unit force applied to the center of mass of a given floor.
26

x

y

θ

x

y

CM

CM

Member 1Member 5

Member 7 Member 8

Member 2

Member 6

Member 3

Member 4

4m 4m

6m

6m

Fx2(t)

Fx1(t)

Fy2(t)

Fy1(t)

Fθ2(t)

θ Fθ1(t)

3. Information relating to the structural loading. In particular HR_DAD_1.1 is capable of

considering:

§ Time varying loads applied at the center of mass of each floor, obtained from wind

tunnel tests carried out on rigid models

§ Static loads applied to each member.

Figure A1.Simple 2- story 3D building.

4. Information relating to the wind climate at or near the building site

§ In general, large separate databases of directional wind speeds are needed for each

type of wind occurring at the site (e.g., hurricanes, thunderstorms, synoptic

winds). The calculation of mean recurrence intervals of wind effects is analogous to

the simple calculations discussed for wind speeds in [17, p. 28]. An algorithm for

incorporating this calculation is currently being developed.

§ Information regarding the ratio between wind speeds at 10 m above ground in open

terrain and their mean hourly wind speed counterparts (as affected by veering, see
27

[17, p. 15] at the top of the building. An algorithm for incorporating the effect of

veering is currently being developed.

A7. Output from Running HR_DAD_1.1

§ Peak demand/capacity ratios for all members calculated for any number of Mean

Recurrence Intervals (MRIs).

§ Peak inter-story drift calculated for any number of column lines and MRIs.

§ Peak top floor acceleration calculated for any number of points and any number of

MRIs.

A8. Details on Use of the Software

The following pages will illustrate in detail how to use the eight pages that make up the GUI of

HR_DAD_1.1. During the general description of the necessary input, and the formatting of such

input, the variables necessary for running the example building shown in Fig. A1 will be input

under the heading 3D 2-story example.
28

PAGE_ONE

INPUT:

H_bldg = Height of the building in meters

3D 2-story example: total building height is 8m.

Nfloors = Number of floors composing the building

3D 2-story example: 2 floors.

Fdofs = Degrees of freedom per floor

3D 2-story example: 3.

No. of modes = Choose the number of vibrational modes to be considered in the analysis

3D 2-story example: 3 modes are chosen.
29

Modal Periods = assign the modal periods in seconds to each vibrational mode

3D 2-story example: if 3 modes are to be considered in the analysis and the

periods are, for example, 1.7 s, 1.5 s and 0.8 s, then the following vector would be

entered:

[0.6122 0.6102 0.4565]

Note: The variable must be input using square brackets as shown.

Modal damping, % = assign the modal damping as a percentage of critical.

3D 2-story example: if 3 modes are to be considered in the analysis and the modal

damping is to 1.5% of critical for each mode, then the following vector should be

entered;

[1.5 1.5 1.5]

Note: The variable must be input using square brackets as shown.

Mode shapes = Input the Matlab mat file containing the mode shapes.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the mode shapes must be named evectors. The file containing the

variable maybe named anyway desired. The variable evectors is a matrix whose

columns contain the mode shapes referred to the center of mass of each floor.

Each column has the x-coordinates first then the y-coordinates and finally the θcoordinates of the mode shape.

3D 2-story example: consider the following mode shapes:

0.62

1

0

mode shape 1

0

0

0

 

 

 

 

=  

 

 

 

 

,

0

0

0.62

mode shape 2

1

0

0

 

 

 

 

=  

 

 

 

 

 and

0

0

0

mode shape 3

0

0.62

1

 

 

 

 

=  

 

 

 

 

.

The variable evectors would therefore take on the following form:
30

0.62 0 0

100

0 0.62 0

010

0 0 0.62

001

 

 

=

 

evectors

and could be saved for example as ModeShapes.mat

END PAGE_ONE INPUT
31

PAGE_TWO

INPUT:

flnFl = load the MATLAB mat file containing the time histories of the floor loads.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the time histories must be named F. The mat file containing the

variable F can then be saved under any name but must end with “_XXX” where

XXX gives the wind direction, in degrees, from which the wind was blowing in

the wind tunnel when the loads where ascertained. An example is shown in Fig. 2

for directions XXX = 000 and XXX = 035.

For each wind direction a separate mat file is needed containing the relative

variable F. The variable F is a matrix, each line of which contains the time history

of a floor load acting in one of the directions x, y or θ. The first N (N is the

number of floors) rows correspond to the floor loads acting in the x-direction

starting from the first floor. The Next N rows correspond to the floor loads acting

in the y-direction while the last N rows correspond to the floor loads acting in the

θ-direction. F will have a total of 3N rows.
32

x

y

XXX=000

XXX=035

35°

θ

Figure A2.File naming convention used for the floor loads.

3D 2-story example: floor loads for running the example can be downloaded from

the site www.nist.gov/wind.

Vm = mean wind speed in m/s at roof of the model during the wind tunnel tests.

3D 2-story example: 20m/s

freq = sampling frequency used in the wind tunnel.

3D 2-story example: 250Hz

ms = scale of the model used in the wind tunnel tests.

3D 2-story example: if the scale was 1/500 then the correct number to input is

500.

Npoints = total number of points that make up the time histories of the floor loads.

3D 2-story example: if 30s of data is recoded during the wind tunnel tests with a

sampling frequency (freq) of 250Hz, then the time histories of the floor loads will

have a total of 7504 points.

Nstart = number of points to be cut from beginning of the time histories during the analysis.

3D 2-story example: 200 points is reasonable.

Note: Numerical integration needs a certain number of points before it stabilizes.

Therefore a certain number of initial points should be cut form the solution of the

equations of motion before estimating the response parameters.

END PAGE_TWO INPUT
33

PAGE_THREE

INPUT:

flnMem = load the list of members composing the structure.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the list of members must be named mem_list. The mat file

containing the variable mem_list can then be saved under any name. The variable

mem_list is a matrix with two rows and number of columns equal to the number

of members that make up the structure. The first row contains the member

numbers while the second row contains a label identifying whether the member is

a beam, column or diagonal. The labels must be a string of characters that begin

with the letters “B” if the element is a Beam, “C” if the element is a column or

“D” if the element is a diagonal.

3D 2-story example: the structure has a total of 8 members. The variable

mem_list would have the following appearance:
34

Y

X

Z

Initial section

Mid section

Terminal section

12345678

"""""""""""""""" CCCCCCCC

 

=    mem_list

and could be saved for example as members_list.mat

flnDif = load the mat file containing the influence coefficients.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the influence coefficients must be named dif. The mat file

containing the variable dif can then be saved under any name. The variable dif is a

3D array. The following description of dif will be in reference to the member

shown in Fig. 3.

Figure A3. ith member of the structure.

Each face of dif contains the influence coefficients associated with the six internal

forces (axial force, shear forces and bending moments in the local X and Y

directions and torsion) that can arise in the initial, mid or terminal sections of a

given member due to a unit force applied to the center of mass of a given floor in

one of the directions x, y or θ. In particular, the first column contains the axial

forces, second and third columns the X and Y shears, fourth column the torsion

and the fifth and sixth columns the X and Y bending moments. The first N rows (N

is the number of floors of the building) are the internal forces in the initial section

due to unit forces applied to the centers of mass of each floor in the x-directions

starting from the 1st floor. The next N rows are due to unit forces in the ydirection while successive N rows are due to unit torques applied to the centers of

mass. This makes up a total of 3N rows for the influence coefficients of the initial

section. The following 6N rows are the influence coefficients for the mid and

terminal sections making a total of 9N rows. The index of each face identifies the

member. Indeed the ith face must correspond to the member described in the ith

column of the variable mem_list.

3D 2-story example: the structure in Fig. A1 has 8 members all of which are

columns. Assume them to be steel tubular members with an outside dimension of

25cm and flange thickness of 6mm. Also assume the steel to have an elastic
35

modulus of 200,000 N/mm2. The next paragraph will explain how to calculate the

first face of the array dif for this example building.

The first face will coincide with the element in the first column of the variable

mem_list. It will therefore coincide with member 1. Because the structure has 2

floors, the first face will have 18 rows. The easiest way to calculate the influence

coefficients is by constructing a model of the building in a commercial software

analysis program, e.g., SAP2000. Within this environment it is then possible to

define 3N = 6 (N is the number of floors which in this case is 2) load cases, one

for each unit force applied to the centers of mass of each floor. By running the

analysis for the 6 load cases and exporting the results in Microsoft Excel format,

tables similar to the one shown in Table A1 are obtained, where the load cases

have been named influx1, influx2,...,influz2. The influence coefficients are

simply the internal forces highlighted in red. To create the MATLAB mat file

simply read the relative rows and columns into MATLAB and construct the array.

Table A1. SAP2000 Excel spreadsheet output for member 1.

TABLE: Element Forces - Frames

Frame Station Output

Case

Case

Type

P V2 V3 T M2 M3

Text m Text Text N N N N-m N-m N-m

1 0 influx1 LinStatic 0 0.25 0 0 0 0.5

1 0 influx2 LinStatic 0 0.25 0 0 0 0.5

1 0 influy1 LinStatic 0 0 0.25 0 0.5 0

1 0 influy2 LinStatic 0 0 0.25 0 0.5 0

1 0 influz1 LinStatic 0 0.039 -0.039 0.011 -0.079 0.079

1 0 influz2 LinStatic 0 0.039 -0.039 0.011 -0.079 0.079

1 2 influx1 LinStatic 0 0.25 0 0 0 0

1 2 influx2 LinStatic 0 0.25 0 0 0 0

1 2 influy1 LinStatic 0 0 0.25 0 0 0

1 2 influy2 LinStatic 0 0 0.25 0 0 0

1 2 influz1 LinStatic 0 0.039 -0.039 0.011 0 0

1 2 influz2 LinStatic 0 0.039 -0.039 0.011 0 0

1 4 influx1 LinStatic 0 0.25 0 0 0 -0.5

1 4 influx2 LinStatic 0 0.25 0 0 0 -0.5

1 4 influy1 LinStatic 0 0 0.25 0 -0.5 0

1 4 influy2 LinStatic 0 0 0.25 0 -0.5 0

1 4 influz1 LinStatic 0 0.039 -0.039 0.011 0.079 -0.079

1 4 influz2 LinStatic 0 0.039 -0.039 0.011 0.079 -0.079

The mat file containing the variable dif can be saved, for example, as dif_all.mat

flnProps = load the mat file containing the section properties.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the list of members must be named props. The mat file containing

the variable props can then be saved under any name. The variable props is a
36

matrix with first column coinciding with the first row of mem_list and therefore

contains the member numbering scheme. Each row contains the strength

capacities of the member identified in the first column of the row. In particular the

second column of each row contains the maximum tensile strength, the third

column the maximum compressive strength while the forth and fifth columns the

bending strengths in the two local X and Y directions.

3D 2-story example: The variable props has the following appearance:

1 1453525 1308172 133024 133024

2 1453525 1308172 133024 133024

3 1453525 1308172 133024 133024

4 1453525 1308172 133024 133024

5 1453525 1308172 133024 13302

props =

4

6 1453525 1308172 133024 133024

7 1453525 1308172 133024 133024

8 1453525 1308172 133024 133024

 

 

 

The mat file containing the variable props could be saved as

frame_properties.mat

WS = wind speeds at roof height of the full scale building for which the response is desired.

3D 2-story example: for example, if 5m/s to 50m/s with 5m/s increments are

desired then the following vector would be entered:

[5 10 15... 45 50]

Note: The variable must be input using square brackets as shown. Please note that

the following notation maybe used [5:5:50].

WD = wind directions for which the response is desired.

3D 2-story example: for example, if 0°, 10°, 20°... 350° and 360° are desired,

then the following vector would be entered:

[0 10 20...350 360]

Note 1: The angles input in WD must be the same as or a subset of the angles for

which the wind loading is known.

Note 2: The variable must be input using square brackets as shown. Please note

that the following notation may be used [0:10:360].
37

MRIBij = The Mean Recurrence Intervals for which the peak demand/capacity ratios are

wanted.

3D 2-story example: for example, if the peak demand/capacity ratios for Mean

Recurrence Intervals of 20 and 50 years are desired, then the following vector

would be entered:

[20 50]

Note: The variable must be input using square brackets as shown.

MRIDrAc = The Mean Recurrence Intervals for which the peak inter-story drift and top floor

acceleration is desired.

3D 2-story example: for example, if the peak inter-story drift and top floor

acceleration for Mean Recurrence Intervals of 10 and 20 years are desired, then

the following vector would be entered:

[10 20]

Note: The variable must be input using square brackets as shown.

intmeth = choose the interpolation method that will be implemented during the calculation of the

responses with given Mean Recurrence Intervals.

Note: The methods will give similar results. However method A will give more

accurate results but take longer than method B, which tends to be more

conservative.

END PAGE_THREE INPUT
38

PAGE_FOUR

INPUT:

flnMass = load the mass and moments of inertia associated with each floor.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the mass and moments of inertia must be named mass. The mat file

containing the variable mass can then be saved under any name. The variable

mass is a column vector. The first three rows are associated with the first floor. In

particular the first entry is the mass in the x-direction, the second entry is the mass

in the y-direction, while the third entry is the moment of inertia with respect to the

center of mass. The next 3 rows are associated with the second floor and so forth.

3D 2-story example: considering for both floors a mass in the x and y directions of

30000kg and moment of inertia of 310000kg m2, the variable mass is:
39

30000

30000

310000

30000

30000

310000

 

 

 

 

=  

 

 

 

 

mass

The mat file containing the variable mass could be saved as mass_asc.mat

flnDLr = load file containing the contribution of the dead weight.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the contribution of the dead weight must be named frames_DL.

The mat file containing the variable frames_DL can then be saved under any

name. The variable frames_DL is a matrix with first column coinciding with the

first row of mem_list and therefore contains the member numbering scheme. Each

row contains the internal forces due to dead weight occurring in the initial, mid

and terminal sections of the member identified in the first column of the row. In

particular columns 2 to 4 contain the axial force and bending moments in the two

local X and Y directions (Fig. A3) for the initial section. Columns 5 to 7 contain

the same information for the mid section, while columns 8 to 10 are dedicated to

axial and bending forces occurring in the terminal section.

3D 2-story example: from a SAP2000 or equivalent model of the building it is

possible to access the internal forces that occur in the sections of interest of the 8

members composing the structure. This will give the following frames_DL:

1 -153606 0 0 -152704 0 0 -151803 0 0

2 -76803 0 0 -75901 0 0 -75000 0 0

3 -153606 0 0 -152704 0 0 -151803 0 0

4 -76803 0 0 -759

frames_DL =

01 0 0 -75000 0 0

5 -153606 0 0 -152704 0 0 -151803 0 0

6 -76803 0 0 -75901 0 0 -75000 0 0

7 -153606 0 0 -152704 0 0 -151803 0 0

8 -76803 0 0 -75901 0 0 -75000 0 0

 

 

 

The mat file containing the variable frames_DL could be saved as

frames_DeadLoad.mat

flnLLr = load file containing the contribution of the live loads.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the contribution of the live loads must be named frames_LL. The
40

mat file containing the variable frames_SDL can then be saved under any name.

The variable frames_SDL is a matrix with first column coinciding with the first

row of mem_list and therefore contains the member numbering scheme. Each row

contains the internal forces due to superimposed dead load occurring in the initial,

mid and terminal sections of the member identified in the first column of the row.

In particular, columns 2 to 4 contain the axial force and bending moments in the

two local X and Y directions (Fig. A3) for the initial section. Columns 5 to 7

contain the same information for the mid section, while columns 8 to 10 are

dedicated to axial and bending forces occurring in the terminal section.

3D 2-story example: considering as superimposed dead load a distributed load of

0.5KN/m2 from the SAP2000 or equivalent model of the building it is possible to

access the internal forces that occur in the sections of interest of the 8 members

composing the structure. This will give the following frames_SDL:

1 -9000 0 0 -9000 0 0 -9000 0 0

2 -4500 0 0 -4500 0 0 -4500 0 0

3 -9000 0 0 -9000 0 0 -9000 0 0

4 -4

frames_SDL =

500 0 0 -4500 0 0 -4500 0 0

5 -9000 0 0 -9000 0 0 -9000 0 0

6 -4500 0 0 -4500 0 0 -4500 0 0

7 -9000 0 0 -9000 0 0 -9000 0 0

8 -4500 0 0 -4500 0 0 -4500 0 0

 

 

 

The mat file containing the variable frames_SDL could be saved as

frames_SDeadLoad.mat

flnSDLr = load file containing the contribution of the superimposed dead load.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the contribution of the superimposed dead load must be named

frames_LL. The mat file containing the variable frames_LL can then be saved

under any name. The variable frames_LL is a matrix with first column coinciding

with the first row of mem_list and therefore contains the member numbering

scheme. Each row contains the internal forces due to live loads occurring in the

initial, mid and terminal sections of the member identified in the first column of

the row. In particular, columns 2 to 4 contain the axial force and bending

moments in the two local X and Y directions (Fig. A3) for the initial section.

Columns 5 to 7 contain the same information for the mid section while columns 8

to 10 are dedicated to axial and bending forces occurring in the terminal section.

3D 2-story example: considering as live load a distributed load of 2KN/m2 from

the SAP2000 or equivalent model of the building it is possible to access the
41

internal forces that occur in the sections of interest of the 8 members composing

the structure. This will give the following frames_LL:

1 -36000 0 0 -36000 0 0 -36000 0 0

2 -18000 0 0 -18000 0 0 -18000 0 0

3 -36000 0 0 -36000 0 0 -36000 0 0

4

frames_LL =

-18000 0 0 -18000 0 0 -18000 0 0

5 -36000 0 0 -36000 0 0 -36000 0 0

6 -18000 0 0 -18000 0 0 -18000 0 0

7 -36000 0 0 -36000 0 0 -36000 0 0

8 -18000 0 0 -18000 0 0 -18000 0 0

 

 

 

The mat file containing the variable frames_LL could be saved as

frames_LiveLoad.mat

DLf = dead load factor.

Note: the dead loads, superimposed dead loads, live loads and wind loads must be

combined. HR_DAD_1.1 allows the user to combine these loads by defining load

factors which are then used to define appropriate load combinations.

3D 2 story example: using the ASCE 7-05 Standard load combination:

1.2D + 1.0L + 1.6W

in which D is the dead and superimposed dead loads, L is the live load while W is

the wind load, the appropriate value is therefore 1.2.

SDLf = superimposed dead load factor.

3D-2 story example: see above under DLf .

LLf = live load factor.

3D-2 story example: see above under DLf.

WLf = wind load factor.

3D-2 story example: see above under DLf.

g = local response peak factor.
42

Note: the local response peak factor is used to calculate the peak internal forces

occurring in the various members. Appropriate values of this parameter are

between 3 and 4.

3D 2 story example: an appropriate value would be 3.5.

END PAGE_ FOUR INPUT
43

PAGE_FIVE

INPUT:

interstory_location = load the file containing the position of the column line where the interstory drift is desired.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the position of the column line must be named

interstory_location. The mat file containing the variable interstory_location can

then be saved under any name. The variable interstory_location is a matrix, the

first 2 rows of which contain the x and y coordinates of the column line with

respect to the center of mass of each floor. The third and last column contains the

height of the story. The first row contains information relating to the first floor,

while the second row stores information related to the second floor and so forth.

Successive column lines are appended as an extra N rows (where N is the number

of floors).

3D 2-story example: wanting to consider the column line formed by members 7-8

and 5-6 (Fig. A1) the variable interstory_location is:
44

334

334

3 34

3 34

 

 

=  

  −

    −

interstory_location

The mat file containing the variable interstory_location could be saved as

Interstory_Drift_Input.mat

Qss = choose whether to consider observed peaks or peaks estimated as indicated in Sect. III B,

www.nist.gov/wind.

Note: the default is to consider the observed peaks of the inter-story drift time

histories.

acceleration_location = load the file containing the positions of the points belonging to the top

floor where the peak acceleration is desired.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the position of the points must be named acceleration_location.

The mat file containing the variable acceleration_location can then be saved

under any name. The variable acceleration_location is a matrix, the first 2 rows

of which contain the x and y coordinates of the point with respect to the center of

mass of the top floor. The first row contains information relating to the first point

while the second row stores information related to the second point and so forth.

3D 2-story example: If one of the corner points is considered, for instance the

point with the coordinates x = 3 and y = 3, the variable acceleration_location is:

acceleration_location = [ ] 3 3

The mat file containing the variable acceleration_location could be saved as

Acceleration_Input.mat

Qs = choose whether to consider observed peaks or peaks estimated as indicated in Sect. III B,

www.nist.gov/wind.

Note: the default is to consider the observed peaks of the top floor acceleration

time histories.

The following pertains to the simulated 999 extreme wind events provided for a large number of

locations (mileposts) along the Gulf of Mexico and North Atlantic coast. This simulated data is

publicly available at http://www.nist.gov/wind by following the links for extreme wind data sets.
45

Hmp = hurricane milepost

3D 2-story example: considering the building located in Miami the milepost is

1450.

flHfile = folder location of the database of simulated hurricanes.

3D 2-story example: after downloading the simulated hurricane database place it

into any folder on your machine and point flHfile to this folder.

Vth = minimum wind speed under which the response is no longer desired

Note: If a low minimum wind speed is considered this will lengthen the

calculation of the responses with a specified MRI. However the higher the

minimum wind speed is the greater the possibility of not considering a critical

wind effect is.

3D 2-story example: a reasonable minimum speed is 15 m/s

Note: In general several types of wind need to be considered – see Sect. 6, item 4

in this report. Software for inclusion of up to three types of wind is being

developed. See also Grigoriu (2008).

END PAGE_ FIVE INPUT
46

PAGE_SIX

OUTPUT:

flnSaveRsBij = specify file location and name where the member response surfaces will be

saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the response surfaces will be named Bij_RS. The mat file

containing the variable Bij_RS can then be saved under any name. The variable

Bij_RS is a 3D array, each face of which represents the response surface of a

specific member. The file structure is schematically represented in Fig. A4.

In particular the index of each face corresponds to the column index of mem_list

and therefore identifies the member. An element of a specific face identified by

the indexes (i,j) corresponds to the peak demand/capacity ratio for the wind

direction given by the ith element of WD and the wind speed given by the jth

element of WS. Therefore each face will have a number of rows coinciding with

the number of wind directions in WD, and a number of columns coinciding with

the number of wind speeds in WS.
47

Members

WD =

WS =

0.024

0.058

0.023

0.057

0.086

0.012

0.056

0.034

0.034

0.087

0.056

0.014

0.034

0.089

0.019

0.067

0.034

0.021

0.067

0.017

0°

90°

180°

270°

360°

20m/s 70m/s 120m/s 170m/s Face 1(member 1)

Face 2 (member 2)

Face 3 (member 3)

Face 4 (member 4)

Face 5 (member 5)

Figure A4. Bij_RS file structure.

3D 2-story example: plotting the first face of Bij_RS gives a graphical

representation of the response surface associated with member 1. The plot is

shown in Fig. A5.

μ

Figure A5.Response surface associated with member 1.
48

Inter-story drift

x-direction

WS =

0.024

0.058

0.023

0.057

0.086

0.012

0.056

0.034

0.034

0.087

0.056

0.014

0.034

0.089

0.019

0.067

0.034

0.021

0.067

0.017

0°

90°

180°

270°

360°

20m/s 70m/s 120m/s 170m/s

Face 1 (story 1)

Face 2 (story 2)

Face N (story N)

Face N+1 (story 1)

Face N+2 (story 2)

Face 2N (story N)

Inter-story drift

y-direction

WD =

flnSaveRsInDr = specify file location and name where the inter-story response surfaces will be

saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the response surfaces will be named InDr_RS_set_X

where X depends on the column number under consideration. The mat file

containing the variable InDr_RS_set_X can then be saved under any name. The

variable InDr_RS_set_X is a 3D array, each face of which represents the

response surface in a specific direction (x or y). In particular the file is arranged so

that the first N faces (N is the number of floors of the building) of the array are

associated with the response in the x-direction while the next N faces are

associated with the y-direction. The structure of the file is schematically shown in

Fig. A6. An element of a specific face identified by the indexes (i,j) corresponds

to the peak inter-story drift for the wind direction given by the ith element of WD

and the wind speed given by the jth element of WS. Therefore each face will have

a number of rows coinciding with the number of wind directions in WD, and a

number of columns coinciding with the number of wind speeds in WS.

Figure A6. InDr_RS_set_X file structure.
49

3D 2-story example: plotting the second face of InDr_RS_set_1 gives a graphical

representation of the response surface associated with the inter-story drift in

direction x of column line 1. The plot is shown in Fig. A7.

Figure A7. 2nd x-direction floor response surface of set 1

flnSaveRsAcc = specify file location and name where the top floor acceleration response

surfaces will be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the response surfaces will be named Acc_RS_point_X

where X depends on the point belonging to the top floor. The mat file containing

the variable Acc_RS_point_X can then be saved under any name. The variable

Acc_RS_point_X is a 3D array, each face of which represents the response

surface in a specific direction (x or y). In particular, the file is arranged so that the

first face of the array is associated with the response in the x-direction while the

next face is associated with the y-direction. The structure of the file is

schematically shown in Fig. A8. As can be seen an element of a specific face

identified by the indexes (i,j) corresponds to the peak inter-story drift for the wind

direction given by the ith element of WD and the wind speed given by the jth

element of WS. Therefore each face will have a number of rows coinciding with

the number of wind directions in WD, and a number of columns coinciding with

the number of wind speeds in WS.
50

WD =

WS =

0.024

0.058

0.023

0.057

0.086

0.012

0.056

0.034

0.034

0.087

0.056

0.014

0.034

0.089

0.019

0.067

0.034

0.021

0.067

0.017

0°

90°

180°

270°

360°

20m/s 70m/s 120m/s 170m/s Face 1 (x-direction)

Face 2 (y-direction)

Figure A8. Acc_RS_point_X file structure.

3D 2-story example: Plotting the faces of Acc_RS_point_1 gives results similar to

those shown for the member response surfaces and inter-story drift.

flnSaveBijMRI = specify file location and name where the member responses with specified

Mean Recurrence Intervals (MRIs) will be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the member responses with specified MRIs will be named

Bij_MRIs. The mat file containing the variable Bij_MRIs can then be saved

under any name. The variable Bij_MRIs is a matrix the rows of which contain the

peak capacity/demand ratios for a specific MRI. In particular the first row will

contain the ratios corresponding to the first MRI contained in MRIBij, the second

row those corresponding to the second MRI of MRIBij and so forth. The index of

each column corresponds to the column index of mem_list and therefore

identifies the member.

3D 2-story example: considering the MRIs input on page 3, MRIBij = [20 50],

Bij_MRIs takes on the following form:

1.11 0.41 1.35 0.49 1.33 0.49 1.18 0.43

1.33 0.49 1.66 0.60 1.62 0.59 1.39 0.51

 

=    Bij_MRIs

Therefore Bij_MRIs(1,1) = 1.11 means the for member 1 the maximum

demand/capacity ratio with a MRI of 20 years is 1.11.

flnSaveInDrMRI = specify file location and name where the inter-story drift responses with

specified Mean Recurrence Intervals (MRIs) will be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the inter-story responses with specified MRIs will be
51

named InDr_MRIs_set_X where X depends on the column number under

consideration. The mat file containing the variable InDr_MRIs_set_X can then

be saved under any name. The variable InDr_MRIs_set_X is a 3D array, each

face of which represents the inter-story drift response, for column line X, with

specified MRI. In particular the first face will contain the responses corresponding

to the first MRI contained in MRIDrAc, the second face those corresponding to

the second MRI of MRIDrAc and so forth. Each face has two columns

corresponding to the x and y directions, respectively, and N rows which

correspond to the N floors.

3D 2-story example: Considering the MRIs input on page 3, MRIDrAc = [10 20],

the first face of InDr_MRIs_set_1 takes on the following form:

0.025 0.026

 ( 1)

0.015 0.016

face  =    InDr_MRIs_set_1

Therefore InDr_MRIs_set_1(1,1, face 1) = 0.025 means that the peak inter-story

drift in direction x of column line 1, floor 1 with MRI of 10 years is 0.025. In

direction y floor 1, it is InDr_MRIs_set_1(1,2, face 1) = 0.026 and so forth.

flnSaveAccMRI = specify file location and name where the top floor acceleration responses

with specified Mean Recurrence Intervals (MRIs) will be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the top floor acceleration responses with specified MRIs

will be named Acc_MRIs_point_X where X depends on the point belonging to

the top floor. The mat file containing the variable Acc_MRIs_point_X can then

be saved under any name. The variable Acc_MRIs_point_X is a 3D array, each

face of which represents the top floor acceleration response, for point X, with

specified MRI. In particular the first face will contain the responses corresponding

to the first MRI contained in MRIDrAc, the second face those corresponding to

the second MRI of MRIDrAc and so forth. Each face has two columns

corresponding to the x and y directions.

3D 2-story example: Considering the MRIs input on page 3, MRIDrAc = [10 20],

the first face of Acc_MRIs_point_1 takes on the following form:

Acc_MRIs_point_1 ( 1) 0.0319 0.0326 face = [ ]

Therefore Acc_MRIs_point_1 (1,1, face 1) = 0.0319 means that the peak top

floor acceleration in direction x of point 1 with MRI of 10 years is 0.0329, while

in direction y it is Acc_MRIs_point_1 (1,2, face 1) = 0.0326 and so forth.

END PAGE_ SIX OUTPUT
52

PAGE_SEVEN (Additional output)

DISPLACEMENTS:

AnGld = choose whether to save select time histories of displacements.

displacement_location = load the file containing the position of the column line where the time

histories of the displacements are desired.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the position of the column line must be named

displacement_location. The mat file containing the variable

displacement_location can then be saved under any name. The variable

displacement_location is a matrix, the first 2 columns of which contain the x and

y coordinates of the column line with respect to the center of mass of each floor.

The first row contains information relating to the first floor while the second row

stores information related to the second floor and so forth. Successive column

lines are appended as an extra N rows (where N is the number of floors).

3D 2-story example: If the column line formed by members 7-8 (Fig. A1) is

considered, the variable displacement_location is:
53

3 3

3 3

 

=    displacement_location

The mat file containing the variable displacement_location could be saved as

Displacement_Input.mat

wsD = wind speeds at roof height of the full scale building for which the displacement time

histories are desired.

3D 2-story example: For example, if 30m/s, 40m/s are desired then the following

vector would be entered:

[30 40]

Note: The variable must be input using square brackets as shown and must be

equal or a subset of WS

wdD = wind directions for which the displacement time histories are desired.

3D 2-story example: For example, if 40° and 180° are desired, then the following

vector would be entered:

[40 180]

Note 1: The variable must be input using square brackets as shown and must be

equal to or a subset of WD

Note 2: The variables wsD and wdD must have the same lengths. Indeed the time

histories of the displacements are calculated for pairs of wind speeds and

directions.

flnSaveDTh = specify file location and name where the time histories of the displacements will

be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the displacements time histories will be named

disp_set_X_wdD_wsD where X depends on the column line while wdD and wsD

will take the values associated with the time history. Therefore for each wind

direction and speed a separate mat file will be generated. The mat file containing

the variable disp_set_X_wdD_wsD can then be saved under any name but will

end with _set_X_wdD_wsD indentifing the column line and wind speed and

direction. The variable disp_set_X_wdD_wsD is a 3D array, each face of which

represents the displacement time histories in a particular direction. The first face is

associated with the x-direction, the second with the y-direction while the third and

last is associated with the rotation around the z-axis. The faces are arranged so that
54

0 1000 2000 3000 4000 5000 6000 7000

0.005

0.01

0.015

0.02

0.025

0.03

Points in the time history

Displacements (m)

each row contains a time history. The first row is associated with the first floor,

the second row with the second floor and so forth.

3D 2-story example: Having considered a single column line (members 1-2, Fig.

A1) with two sets of wind speeds and directions, two mat files will be saved

containing the variables disp_set_1_40_30 and disp_set_1_180_40. If it is

desired to display the time histories associated with the displacements in the xdirection at the top of column line 1, for a wind speed of 40m/s and direction of

30°, it is simply necessary to plot disp_set_1_40_30(2,:,1); this constitutes the

second row of the first face. The time history is shown in Fig. A9.

Figure A9. disp_set_1_40_30(2,:,1), 2nd floor x-direction displacement time history for column

line 1.

TOP FLOOR ACCELERATION:

AnGla = choose whether to save select time histories of top floor accelerations.

acc_location = load the file containing the positions of the points where the time histories of the

top floor accelerations are desired.

File structure: The file can be constructed in MATLAB. The variable in the mat

file containing the position of the points must be named acc_location. The mat

file containing the variable acc_location can then be saved under any name. The

variable acc_location is a matrix, the first 2 columns of which contain the x and y

coordinates of the points with respect to the center of mass of the top floor. The
55

first row contains information relating to the first point while the second row

stores information related to a possible second point and so forth.

3D 2-story example: If the point above member 8 (Fig. A1) is considered, the

variable acc_location is:

acc_location = [ ] 3 3

The mat file containing the variable acc_location could be saved as

Acc_Input.mat

wsA = wind speeds at roof height of the full scale building for which the top floor acceleration

time histories are desired.

3D 2-story example: For example, if 35m/s, 45m/s are desired then the following

vector would be entered:

[35 45]

Note: The variable must be input using square brackets as shown and must be

equal to or a subset of WS

wdA = wind directions for which the top floor acceleration time histories are desired.

3D 2-story example: For example, if 70° and 250° are desired, then the following

vector would be entered:

[70 250]

Note 1: The variable must be input using square brackets as shown and must be

equal to or a subset of WD

Note 2: The variables wsA and wdA must have the same lengths. Indeed the time

histories of the displacements are calculated for pairs of wind speeds and

directions.

flnSaveATh = specify file location and name where the time histories of the top floor

accelerations will be saved.

File structure: The file that will be saved is a MATLAB mat file. The variable in

the mat file containing the top floor acceleration time histories will be named

Top_floor_acc_wdD_wsD where wdD and wsD will take the values associated

with the particular time history. Therefore for each wind direction and speed a

separate mat file will be generated. The mat file containing the variable

Top_floor_acc_wdD_wsD can then be saved under any name but will end with

_wdD_wsD indentifing the wind speed and direction. The variable
56

Top_floor_acc_wdD_wsD is a 3D array, each face of which represents the top

floor acceleration time histories in a particular direction. The first face is

associated with the x-direction, the second with the y-direction while the third and

last is associated with the rotation around the z-axis. The faces are arranged so that

each row contains a time history. The first row is associated with the first point

input form acc_location, the second row with a possible second point, and so

forth.

3D 2-story example: wanting to evaluate the time history of the top floor

acceleration in direction x of the point input in acc_location for a wind direction

70° and speed 35m/s, it is simply necessary to evaluate

Top_floor_acc_70_35(1,:,1).

END PAGE_ SEVEN
57

PAGE _EIGHT (Additional output)

DISPLACEMENTS:

flnSAVE = specify file location and name where the all the input information loaded through

pages one to seven will be saved.

File structure: The file is a MATLAB mat file that contains all the necessary

information input at the time the file was saved

flnLOAD = load a file containing input information for running HR_DAD_1.1.

File structure: The file is a MATLAB mat file generated during the definition of

flnSAVE.

END PAGE_EIGHT
58