Comparing Base Shear Forces and Displacements of … · ABSTRACT Comparing Base Shear Forces and...

Comparing Base Shear Forces and Displacements of

SDOF and MDOF Models Subjected to

Earthquake Ground Motions

Reed E. Crosby

A project submitted to the faculty of Brigham Young University

in partial fulfillment of the requirements for the degree of

Master of Science

Paul William Richards, Chair Fernando S. Fonseca

Kevin W. Franke

Department of Civil Engineering

Brigham Young University

April 2013

Copyright © 2013 Reed Crosby

All Rights Reserved

ABSTRACT

Comparing Base Shear Forces and Displacements of SDOF and MDOF Models Subjected to

Earthquake Ground Motions

Reed Crosby Department of Civil Engineering, BYU

Master of Science

This project compares base shear and displacement response spectra for single degree of freedom models and multi-degree of freedom models that have varying damping values and stiffness distribution methods. The response spectra are affected by the Rayleigh damping values and further research needs to be performed to generate results that can be validated by real-world tests. The results contained in the project correctly represent the mathematical models but the models may not represent real-life structures. Keywords: SDOF models, MDOF models, OpenSees, base shear Force, Displacement, Rayleigh Damping, Response Spectra.

ACKNOWLEDGEMENTS

Special thanks to Dr. Paul Richards for the guidance given throughout the project and the

many hours spent helping to solve problems encountered throughout the research. Also, thanks

to Kyle Atwood for helping with the stiffness distribution files.

v

TABLE OF CONTENTS

LIST OF TABLES .................................................................................................................................... vii

LIST OF FIGURES ................................................................................................................................... ix

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

2 Methods ............................................................................................................................................... 2

2.1 Creating a Basic Model ............................................................................................................... 2

2.2 Model Parameters ........................................................................................................................ 4

2.3 Earthquake Ground Motions ....................................................................................................... 5

2.4 Outputs ........................................................................................................................................ 7

3 Results and discussion ........................................................................................................................ 9

3.1 Verification .................................................................................................................................. 9

3.1.1 SeismoSoft .............................................................................................................................. 9

3.1.2 PEER Database ..................................................................................................................... 10

3.2 Base Shear ................................................................................................................................. 11

3.2.1 Rayleigh Damping ................................................................................................................. 14

3.3 Displacement ............................................................................................................................. 17

3.4 Conclusions ............................................................................................................................... 20

REFERENCES ........................................................................................................................................... 21

Appendix A. ............................................................................................................................................... 23

A.1 Graphs ....................................................................................................................................... 24

B.4 Stiffness Distribution Values ..................................................................................................... 34

Appendix B. TCL code ........................................................................................................................ 35

B.1 Main Code ................................................................................................................................. 35

B.2 Input File ................................................................................................................................... 38

B.3 Procedures ................................................................................................................................. 40

vii

LIST OF TABLES

Table A-1: Earthquake group A and group B with the file names ........................................23

Table B-2: Stiffness Distribution Values used for Method 2 ................................................34

ix

LIST OF FIGURES

Figure 2-1: Example of a simple model .............................................................................................................3

Figure 2-2: Spectral acceleration response spectra for group A ground motions including the mean spectrum ...............................................................................................................................6

Figure 2-3: Spectral acceleration response spectra for group B ground motions including the mean spectrum. ..............................................................................................................................7

Figure 3-1: Mean Response Spectra from (a) SeismoSpect, (b) SDOF models, (c) 2DOF Models, and (d) 3DOF Models Using Earthquake Group A and 0% Damping ..................................10

Figure 3-2: Comparison of accelerations from the PEER ground motion database (left) and from OpenSees code (right) corresponding to 2% damping .........................................................11

Figure 3-3: Comparison of displacements from the PEER ground motion database (left) and from OpenSees code (right) corresponding to 2% damping .........................................................11

Figure 3-4: Normalized base shear for earthquake group A with 2% damping and all members with uniform stiffness ..........................................................................................................12

Figure 3-5: Normalized base shear for earthquake group A with 0% damping and all members with uniform stiffness ..........................................................................................................12

Figure 3-6: Normalized base shear for earthquake group A with 0% damping and all members with non-uniform stiffness ...................................................................................................13

Figure 3-7: Normalized base shear for earthquake group A with 2% damping and all members with non-uniform stiffness ...................................................................................................13

Figure 3-8: Mass-damping and stiffness-damping components to Rayleigh Damping .....................................14

Figure 3-9: Case I – Rayleigh damping with 5% damping ................................................................................16

Figure 3-10: Case II – Mass-proportional damping only with 5% damping .....................................................16

Figure 3-11: Case III – Stiffness-proportional damping only with 5% damping ...............................................17

Figure 3-12: Displacement for earthquake group A with 0% damping and all members with uniform stiffness ..........................................................................................................................18

Figure 3-13: Displacement for earthquake group A with 2% damping and all members with uniform stiffness ..........................................................................................................................18

Figure 3-14: Displacement for earthquake group A with 2% damping and all members with non-uniform stiffness ..................................................................................................................19

Figure 3-15: Displacement for earthquake group A with 0% damping and all members with non-uniform stiffness ..................................................................................................................19

x

Figure A-1: Stiffness distribution alternative method ....................................................................................... 23

Figure A-2: Normalized base shear for earthquake group A with 5% damping and all members with uniform stiffness ......................................................................................................... 24

Figure A-3: Normalized base shear for earthquake group A with 5% damping and all members with non-uniform stiffness .................................................................................................. 24

Figure A-4: Displacement for earthquake group A with 5% damping and all members with uniform stiffness ......................................................................................................................... 25

Figure A-5: Displacement for earthquake group A with 5% damping and all members with non-uniform stiffness .................................................................................................................. 25

Figure A-6: Normalized base shear for earthquake group B with 0% damping and all members with uniform stiffness ......................................................................................................... 26



Figure A-9: Normalized base shear for earthquake group B with 2% damping and all members with non-uniform stiffness .................................................................................................. 28



Figure A-12: Displacement for earthquake group B with 2% damping and all members with uniform stiffness ......................................................................................................................... 30



Figure A-15: Displacement for earthquake group B with 2% damping and all members with non-uniform stiffness .................................................................................................................. 32



1

1 INTRODUCTION

The equivalent lateral force method uses the natural period from simple, single degree of freedom

(SDOF) models to calculate base shear forces used in seismic design of buildings. The buildings are

represented by “equivalent” SDOF models. The natural period of these simplified models is based on the

height of the building and the type of lateral resisting system used in the building. The period is used to

determine the magnitude of the base shear force by using spectral acceleration graphs and equations.

The spectral acceleration graphs and equations in ASCE 7-10 are all based on SDOF models because

they are easy to analyze compared to multi-degree of freedom (MDOF) models and can lead to a

conservative building design. The SDOF models can be overly conservative though, and result in

unnecessarily high base shear forces. The high base shear forces may result in less economical buildings.

SDOF and MDOF models were created and analyzed in order to discover if the “equivalent” SDOF

systems currently used in design could be improved in order to create a more accurate design method.

The models consisted of masses attached by springs and subjected to earthquake ground motions. The

models create systems that use mathematical means to analyze “buildings” and calculate the base shear

and maximum displacements for a given ground motion.

2

2 METHODS

Models were generated and subjected to ground motions. Base shear forces and

maximum displacements were recorded.

2.1 Creating a Basic Model

Open System for Earthquake Engineering Simulation (OpenSees) software that

incorporates Tool Command Language (Tcl) was used to build and analyze hundreds of models.

OpenSees is open-source software developed by the University of California, Berkley for

“simulating the response of structural and geotechnical systems subjected to earthquakes.” The

OpenSees engine has a number of built-in commands that facilitate the process of creating

models, subjecting them to ground motions, and recording results. The models were built by

writing Tool Command Language (Tcl) scripts that ran in the OpenSees engine. All of the Tcl

scripts used can be seen in Appendix B.

The models consisted of a series of nodes that were connected by elements that were

made up of materials. The nodes were assigned coordinates, mass values, and degrees of

freedom. The first node was fixed and all other nodes were free to move in one direction. Each

node was assigned an equal mass in order to create a uniform mass distribution. The nodes were

then joined by elements that were assigned a stiffness value (k), as seen in Figure 2-1.

T

model wi

all of the

througho

the equiv

decreases

referred t

In

were gen

achieved

eigenvalu

compared

natural p

V

OpenSee

within th

proportio

The relative

ith a desired

elements ha

out the text. M

valent lateral

s. The stiffne

to as “non-u

n the overall

nerated. The

d. The model

ue command

d to the desi

eriod of the

Various Rayl

es provides a

he structure.

onal damping

stiffness of e

d natural peri

ad an equal s

Method two

l force. The s

ess follows a

niform distr

l procedure,

model’s stif

l was built th

d was used to

red natural p

model was t

eigh dampin

a Rayleigh D

Rayleigh da

g using the f

each elemen

iod. Method

stiffness. Th

used a stiffn

stiffness dec

a quadratic c

ibution” thro

explained in

ffness was ch

he first time

o find the fir

period of the

the same as t

ng values we

Damping com

amping comb

following eq

Figure 2-1:

3

nt was varied

one used a u

his will be ref

ness distribu

creases as the

curve from 1

oughout the

n the next sec

hanged until

with a trial s

rst natural pe

e model and

the desired n

ere used and

mmand that w

bines mass-p

quations:

Example of a

d using two d

uniform stiff

ferred to as “

ution based o

e lateral forc

1 to zero (Fig

text.

ction, model

l the desired

stiffness assi

eriod of the m

the stiffness

natural perio

compared th

was used to r

proportional

a simple model

different me

ffness distrib

“uniform dis

on a triangula

ce applied to

gure A-1). Th

ls with differ

natural freq

igned to each

model. The

s was change

od.

hroughout th

represent the

damping wi

l

thods to crea

bution in whi

stribution”

ar distributio

o the element

his will be

rent periods

quency was

h element. T

actual value

ed until the

he analyses.

e damping

ith stiffness-

ate a

ich

on of

t

The

was

-

Damping

values of

for mode

2.2 Mo

T

file that u

T

B.2). The

maximum

number o

method,

g based on th

f 0%, 2%, an

els representi

odel Param

The program

used the othe

The models w

e variables th

m period for

of DOFs, end

and the file n

he first and th

nd 5% damp

ing building

eters

consisted of

er files to pe

were created

hat are defin

the analyses

ding number

names for th

hird modes w

ing. These d

s.

f an input fil

erform the an

based off of

ned in the file

s, the change

r of DOFs, d

he output file

4

was used for

damping valu

le, a procedu

nalyses and c

f variables d

e are mass, s

e of the perio

desired param

es.

r the analyse

ues are the m

ures file, an e

create the ou

defined withi

starting stiffn

od between

meter to reco

es of the MD

most commo

earthquake f

utput file.

in the input f

fness, minim

analyses, da

ord, stiffness

(

(

(

DOF models

only used val

file, and a m

file (Append

mum and

amping, start

s distribution

(2-1)

(2-2)

(2-3)

with

lues

ain

dix

ting

n

5

Procedures were written in the procedures file (Appendix B.3) and called in the main

script (Appendix B.1) in order to break the required actions into smaller parts. Each procedure

requires inputs and then returns an output needed for the main program. The procedures contain

action scripts that build the model, run the earthquake ground motion, record the maximum

outputs, change the starting parameters, and perform other necessary actions to analyze the

models.

The earthquake file is a list of all of the names of the ground motion files that would be

used in the analyses. The ground motion acceleration files were all kept in the same folder to be

used by the program. The list of file names can be seen in Table A-1.

The main script used the above files as well as three main loops to create the output file.

The program created a model and ran it through all of the desired earthquakes (loop 1) read from

a file with a list of the earthquake files that were to be included in the analyses. The model was

then modified to have a different desired natural period. The first loop was repeated for the new

model and the model was modified again. This is repeated until the maximum desired natural

period was reached (loop 2). The number of degrees of freedom was increased and loop 1 and

loop 2 were repeated. The number of DOFs was increased in this manner until the maximum

number of DOFs was reached (loop 3).

2.3 Earthquake Ground Motions

The Pacific Earthquake Engineering Research Center (PEER) Ground Motion Database

was used to create two groups of similar earthquake ground motions. The database was searched

by using a target design spectrum to find two groups with 20 earthquakes each that were

comparable to a target design spectrum. Earthquake sites for Group A include Parkfield, Imperial

Valley, Irpinia (Italy), Morgan Hill, Loma Prieta, Kocaeli (Turkey), Chi-Chi (Taiwan), and

Duzce (T

Westmor

Mendoci

B

The indiv

Figure 2-

A earthqu

used wer

Figure the mea

Turkey). Eart

rland, Coalin

ino.

Both groups o

vidual spectr

-2 and Figur

uakes maint

re unscaled g

2-2: Spectraan spectrum

thquake site

nga, Northrid

of earthquak

ral accelerati

re 2-3 for ear

ain the maxi

ground motio

l acceleration

s for Group

dge, San Fer

kes had simil

ions as well

rthquake gro

imum accele

ons obtained

n response sp

6

B include M

rnando, Fruil

lar unscaled,

as the mean

oup A and ea

eration longe

d from the PE

pectra for gr

Managua (Ni

li (Italy), Im

, maximum m

n spectral acc

arthquake gr

er than group

EER ground

roup A groun

caragua), Li

mperial Valle

mean accele

celeration ca

roup B, respe

p B. All of th

d motion data

nd motions in

ivermore,

ey, and Cape

eration spect

an be seen in

ectively. Gro

he earthquak

abase.

ncluding

e

tra.

n

oup

kes

2.4 Ou

Op

under the

earthquak

found by

order to p

displacem

T

user inpu

the displa

period. T

displacem

values ar

maximum

Figthe

utputs

penSees was

e specified e

kes were ave

y taking the t

provide com

ment was rec

The program

uts and was u

acement is a

The model is

ment for each

re arithmetic

m average di

gure 2-3: Spee mean spectr

used to mea

arthquake gr

eraged to fin

total base she

mparable data

corded for th

output file w

used to creat

as follows: F

then subject

h of the grou

ally average

isplacement.

ectral accelerrum.

asure the bas

round motio

nd an averag

ear reaction

a between SD

he last node i

was created t

te response s

irst, a model

ted to a serie

und motions

ed to produce

. The desired

ration respon

7

se shear and

ons. The max

e maximum

and dividing

DOF models

in the model

through a se

spectra. The

l is built in t

es of ground

s is recorded

e a single po

d natural per

nse spectra fo

maximum d

ximum desir

value. The n

g it by the to

s and MDOF

l.

eries of writt

process to o

the program

d motions an

and stored i

oint correspo

riod is then i

or group B g

displacement

red values fo

normalized b

otal weight o

F models. Th

ten procedur

obtain respon

with the des

nd the maxim

in a list. The

onding to a p

increased an

ground motio

t of each mo

or a group of

base shear w

of the model

he maximum

es with give

nse spectra f

sired natural

mum

ese maximum

period and

d this proces

ons including

odel

f

was

in

m

en

for

m

ss is

g

8

repeated until the maximum specified period is reached and each maximum average

displacement is stored in a list. The number of degrees of freedom is then increased by one and

the steps are all repeated. This whole process is repeated until the maximum specified number of

DOFs is reached. The list for each DOF is printed to an output file that can easily produce

response spectra.

Damping values and the method used to distribute the stiffness were altered to create base

shear and displacement response spectra that could be compared. The normalized base shear and

maximum displacement for each stiffness distribution method with each value of damping were

recorded for both groups of earthquakes. Response spectra were created for each variation.

9

3 RESULTS AND DISCUSSION

3.1 Verification

The analytical models are only valid if the results can be verified and the procedures and

processes used can be validated. Individual procedures as well as the groups of procedures

within the code had to be verified to work correctly and give expected results. The complexity of

MDOF models makes it necessary to verify simple models and then assume that the models

continue to represent more complex models. The methods discussed below were used to verify

individual procedures as well as the code as a whole.

3.1.1 SeismoSoft

SeismoSoft provides earthquake engineering software that analyzes ground motions. The

ground motions can be input into SeismoSoft to create design spectra, find displacements, and

find accelerations for SDOF systems. The values found from SeismoSoft software were

compared to the analytical models. The SeismoSoft data can be seen in Figure 3-1 (a).

Multiple cases had to be compared to make sure that the program was working for each

different case. Figure 3-1 (b) shows results from a simple SDOF model with a single mass on top.

Figure 3-1 (c) shows results from a 2DOF model with a very small mass at the first node and a

large mass at the second (top) node. Figure 3-1 (d) shows the results from a 3DOF model with

FigM

very sma

three DO

structure

response

3.1.2 P

Th

contains

accelerat

compared

Figure 3-2

models w

gure 3-1: MeModels Using E

all masses at

OF models ac

. Earthquake

spectrum fo

PEER Datab

e PEER Gro

response spe

tion and disp

d to the resp

2 and Figure

with 2% dam

ean ResponseEarthquake

the first two

ct just like a

e group A w

or each case

base

ound Motion

ectra results

placement re

ponse spectra

3-3. The res

mping and un

(c)

(a)

e Spectra froGroup A an

o nodes and

SDOF mode

ith 0% damp

is identical,

n Database u

for multiple

sponse spect

a obtained fr

ponse spectr

niform stiffne

om (a) Seismd 0% Dampi

10

a large mass

el because o

ping and uni

validating th

sed to obtain

e damping va

tra for SDOF

rom the writt

ra below cor

ess. This val

oSpect, (b) Sing

s at the third

of the small m

iform stiffne

he program

n all of the e

alues for eac

F systems fr

ten program

rrespond to t

lidates the S

SDOF model

d (top) node.

masses in the

ess was used

when 0% da

earthquake g

ch ground m

rom the data

m and were co

the results fr

DOF model

(

(

s, (c) 2DOF M

The two and

e middle of t

d in all cases.

amping is us

ground motio

motion. The

abase were

ompared in

rom SDOF

s with damp

d)

(b)

Models, and

d

the

. The

sed.

ons

ping.

(d) 3DOF

Figure 3-2OpenSees

3.2 Ba

Re

distributi

method a

affects th

distributi

Results f

Figure 3-from Ope

: Comparisocode (right)

ase Shear

sponse spec

ion were cr

and the damp

he base shea

ion with 0%

for structures

-3: ComparisenSees code

on of acceleracorrespondin

ctra for the n

eated for m

ping in the s

ar values. Th

% and 2% d

s with non-u

son of displa(right) corre

ations from tng to 2% dam

normalized b

models rangi

structure we

he normalize

amping can

uniform stiffn

cements fromesponding to

11

the PEER grmping

base shear fo

ing from 1

ere changed i

ed base shea

be seen in

ness are in F

m the PEER 2% damping

round motion

force with va

to 25 DOF

in order to c

ar for structu

Figure 3-5

Figure 3-6 and

ground motig

n database (l

arying damp

Fs. The stiff

compare how

ures with a

and Figure 3

d Figure 3-7.

ion database

eft) and from

ping and stif

fness distrib

w each param

uniform stif

3-4, respecti

e (left) and

m

ffness

bution

meter

ffness

ively.

12

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-4: Normalized base shear for earthquake group A with 2% damping and all members with uniform stiffness

Figure 3-5: Normalized base shear for earthquake group A with 0% damping and all members with uniform stiffness

13

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-7: Normalized base shear for earthquake group A with 2% damping and all members with non-uniform stiffness

Figure 3-6: Normalized base shear for earthquake group A with 0% damping and all members with non-uniform stiffness

14

The damping used in the structure has a huge effect on the base shear force. The damping

affects the normalized base shear forces of the models with natural periods ranging from 0

seconds to 1.5 seconds the most. The largest change in base shear force is the jump from a SDOF

to a 2DOF model. The 2DOF model has a base shear force ranging from 30%-60% of the SDOF

value from 0.5 seconds to 1.5 seconds. The results show that a SDOF “equivalent” model would

be vastly overconservative for 2% damping if used for a MDOF structure.

The stiffness distribution has little to no effect on the values of the base shear forces. The

base shear force depends on the natural period of the model. The model with uniform stiffness

has to have the same natural period as a model with non-uniform stiffness. The overall stiffness

of the building is what affects the base shear force of the building.

3.2.1 Rayleigh Damping

The Rayleigh damping used is commonly used in modeling structures. It involves a mass-

proportional damping coefficient and a stiffness-proportional damping coefficient (Figure 3-8).

Figure 3-8: Mass-damping and stiffness-damping components to Rayleigh Damping

The adverse affects of Rayleigh damping has been the topic of research for John F. Hall

and Finley A. Charney. Research performed by John F. Hall on damping looks at mis-using

Rayleigh damping. His research shows that the damping force grows unrealistically large in

situations where non-linear analyses are used and further research is needed to formulate better

Dam

ping ζn

Natural Frequency, ωn

Mass Damping

Rayleigh Damping

Stiffness Damping

strategies

in inelast

O

coefficien

different

and equa

damping

w

N

case, the

Each cas

coefficien

damping

shows th

damping

mass-pro

proportio

s to remedy

tic systems.

One remedy

nt and limiti

types of dam

ations descr

were calcul

where the var

Normalized b

mass-propo

e used a targ

nts based o

coefficient

he results us

, and Figure

oportional d

onal damping

the problem

suggested

ing the stiffn

mping on m

ribed above

lated based o

riables are as

base shear f

ortional damp

get damping

on the first

based on th

ing tradition

e 3-11 shows

damping con

g was not lim

ms. Mr. Charn

by Hall req

ness coeffici

models. The R

e. The mass

on the follow

s described i

force respon

ping only ca

g value of 5%

and third n

he first natu

nal Rayleigh

s the stiffne

ntributes th

mited in this

15

ney focuses

quired elim

ient. An ana

Rayleigh dam

s proportion

wing two equ

in section 2.

nse spectra w

ase, and the s

% damping. C

natural frequ

ural frequenc

h damping, F

ss-proportio

he most to

analysis.

on the prob

minating the

alyisis was p

mping was c

nal damping

uations:

1.

were created

stiffness pro

Case I, Rayl

uencies. Ca

cy using the

Figure 3-10 s

onal damping

the Raylei

blems with R

mass-propo

performed to

calculated u

g and stiffn

d for the R

oportional da

leigh dampin

se II and I

e above equa

shows the m

g. The resul

igh dampin

Rayleigh dam

ortional dam

o see the effe

using the met

ness proport

Rayleigh dam

amping only

ng, calculate

II calculated

ations. Figur

mass-proport

lts show tha

g. The stif

(3-1)

(3-2)

mping

mping

ect of

thods

tional

mping

case.

ed the

d the

re 3-9

tional

at the

ffness

16

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-9: Case I – Rayleigh damping with 5% damping

Figure 3-10: Case II – Mass-proportional damping only with 5% damping

17

3.3 Displacement

The displacement response spectra for models representing 1 to 25 story buildings were

also analyzed with the same changing parameters. The displacements for structures with a

uniform stiffness distribution with 0% and 2% damping can be seen in Figure 3-12 and Figure 3-13

and for structures with non-uniform stiffness distribution in Figure 3-14 and Figure 3-15.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-11: Case III – Stiffness-proportional damping only with 5% damping

18

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-13: Displacement for earthquake group A with 2% damping and all members with uniform stiffness

Figure 3-12: Displacement for earthquake group A with 0% damping and all members with uniform stiffness

19

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure 3-14: Displacement for earthquake group A with 2% damping and all members with non-uniform stiffness

Figure 3-15: Displacement for earthquake group A with 0% damping and all members with non-uniform stiffness

20

The damping value affected the displacement in the models. The damping value affected

the overall magnitude of the displacements for all the DOFs equally. The difference between the

DOFs remained similar with increasing damping.

The stiffness distribution method used affected the overall magnitudes of the

displacements as well as the difference between the DOFs. The uniform stiffness distribution

experienced a 40% increase in displacement at six seconds from a SDOF model to a 25 DOF

model. The non-uniform stiffness distribution experienced an 82% increase in displacement at

six seconds from the SDOF model to the 25 DOF model. The increase in displacement is due to

the decreased stiffness of the members at the top of the structure.

All of the results shown above were obtained from earthquake group A. The results

obtained from earthquake group B and also the results with models that have 5% damping can be

seen in Appendix A. They were not included in the main text because the 5% damping gave very

similar results to 2% damping and earthquake group B gave very similar results to earthquake

group A.

3.4 Conclusions

The results correctly represent the mathematical models within the program but the

mathematical may not represent the real-life structures. The models are performing as expected

given the inputs of the program. The effect that Rayleigh damping has on the models suggests

that the damping coefficient is unrealistically large and the results are non-conservative. The

values cannot be used to correctly analyze if the “equivalent” SDOF systems could be improved

in order to create a more accurate design method without further research into the problems

associated with Rayleigh damping.

21

REFERENCES

Charney, Finley A. "Unintended Consequences of Modeling Damping in Structures.(Author Abstract)(Technical Report)." Journal of Structural Engineering 2008: 581.

Hall, John F. "Problems Encountered from the use (Or Misuse) of Rayleigh Damping." Earthquake Engineering & Structural Dynamics 2006: 525-45.

OpenSees, The Regents of the University of California (2006). Retrieved from http://opensees.berkeley.edu/

PEER Ground Motion Database, The Regents of the University of California, Berkeley (2010). Retrieved from http://peer.berkeley.edu/peer_ground_motion_database

Earth

Imperia

Imperia

Imperia

Chi‐Ch

Chi‐Ch

Duzce,

Imperia

Loma P

Kocael

Chi‐Ch

APPEND

T

stiffness

hquake Site

Earthqu

al Valley, Ca

al Valley, Ca

al Valley, Ca

i, Taiwan

i, Taiwan

Turkey

al Valley, Ca

Prieta, Ca

i, Turkey

i, Taiwan

Table

DIX A.

This appendi

distribution

Fig

File N

NGA_no_179_

NGA_no_179_

NGA_no_180_

NGA_no_180_

NGA_no_183_

NGA_no_183_

NGA_no_184_

NGA_no_184_

NGA_no_779_

NGA_no_779_

NGA_no_1158

NGA_no_1158

NGA_no_1504

NGA_no_1504

NGA_no_1508

NGA_no_1508

NGA_no_1511

NGA_no_1511

NGA_no_1605

NGA_no_1605

uake Group A

e A-1: Earthq

ix contains

method used

gure A-1: Stif

Name

_H‐E04140

_H‐E04230

_H‐E05140

_H‐E05230

_H‐E08140

_H‐E08230

_H‐EDA270

_H‐EDA360

_LGP000

_LGP090

8_DZC180

8_DZC270

4_TCU067‐E

4_TCU067‐N

8_TCU072‐E

8_TCU072‐N

1_TCU076‐E

1_TCU076‐N

5_DZC180

5_DZC270

A

quake group

extra inform

d for method

ffness distrib

23

Ea

Man

Live

We

Nor

San

Fuil

Imp

Cap

Coa

Coa

p A and grou

mation abou

d 2 and Tabl

bution altern

arthquake Si

nagua, Nicarag

ermore, Ca

stomorland, C

rthridge, Ca

Fernando, Ca

i, Italy

perial Valley, C

pe Mendocino,

Eart

alinga, Ca

alinga, Ca

up B with the

ut the resea

le A.1 lists th

ative method

ite F

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

NGA_no_

gua

Ca

Ca

, Ca

hquake Grou

e file names

arch. Figure

he earthquak

d

ile Name

_95_A‐MAN09

_95_A‐MAN18

_215_A‐SRM07

_215_A‐SRM34

_315_NIL000

_315_NIL090

_411_D‐PVP27

_411_D‐PVP36

_412_D‐PVY04

_412_D‐PVY13

_983_0655‐022

_983_0655‐292

_77_PUL164

_77_PUL254

_126_GAZ000

_126_GAZ090

_723_B‐PTS225

_723_B‐PTS315

_828_PET000

_828_PET090

up B

e A.1 show

ke groups us

90

80

70

40

70

60

45

35

2

2

5

5

ws the

sed.

24

Figure A-3: Normalized base shear for earthquake group A with 5% damping and all members with non-uniform stiffness

Figure A-2: Normalized base shear for earthquake group A with 5% damping and all members with uniform stiffness

A.1 Graphs

Graphs for a limited number of tests are contained in the main body of this paper. The

graphs for the other analyses are contained below for normalized base shear and displacement.

The first group of graphs is for earthquake group A and the second group of graphs is for

earthquake group B.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

25

Figure A-5: Displacement for earthquake group A with 5% damping and all members with non-uniform stiffness

Figure A-4: Displacement for earthquake group A with 5% damping and all members with uniform stiffness

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

26

Earthquake Group B

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Reaction

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure A-6: Normalized base shear for earthquake group B with 0% damping and all members with uniform stiffness

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


27

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


28

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure A-9: Normalized base shear for earthquake group B with 2% damping and all members with non-uniform stiffness

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


29

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Norm

alized Base Shear

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


30

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure A-12: Displacement for earthquake group B with 2% damping and all members with uniform stiffness

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


31

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


32

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF

Figure A-15: Displacement for earthquake group B with 2% damping and all members with non-uniform stiffness

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


33

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9

Displacement [in]

Period [sec]

1 DOF

2 DOF

5 DOF

15 DOF

25 DOF


34

1.00

1.00

0.75

1.00

0.89

0.56

1.00

0.94

0.75

0.44

1.00

0.96

0.84

0.64

0.36

1.00

0.97

0.89

0.75

0.56

0.31

1.00

0.98

0.92

0.82

0.67

0.49

0.27

1.00

0.98

0.94

0.86

0.75

0.61

0.44

0.23

1.00

0.99

0.95

0.89

0.80

0.69

0.56

0.40

0.21

1.00

0.99

0.96

0.91

0.84

0.75

0.64

0.51

0.36

0.19

1.00

0.99

0.97

0.93

0.87

0.79

0.70

0.60

0.47

0.33

0.17

1.00

0.99

0.97

0.94

0.89

0.83

0.75

0.66

0.56

0.44

0.31

0.16

1.00

0.99

0.98

0.95

0.91

0.85

0.79

0.71

0.62

0.52

0.41

0.28

0.15

1.00

0.99

0.98

0.95

0.92

0.87

0.82

0.75

0.67

0.59

0.49

0.38

0.27

0.14

1.00

1.00

0.98

0.96

0.93

0.89

0.84

0.78

0.72

0.64

0.56

0.46

0.36

0.25

0.13

1.00

1.00

0.98

0.96

0.94

0.90

0.86

0.81

0.75

0.68

0.61

0.53

0.44

0.34

0.23

0.12

1.00

1.00

0.99

0.97

0.94

0.91

0.88

0.83

0.78

0.72

0.65

0.58

0.50

0.42

0.32

0.22

0.11

1.00

1.00

0.99

0.97

0.95

0.92

0.89

0.85

0.80

0.75

0.69

0.63

0.56

0.48

0.40

0.31

0.21

0.11

1.00

1.00

0.99

0.98

0.96

0.93

0.90

0.86

0.82

0.78

0.72

0.66

0.60

0.53

0.46

0.38

0.29

0.20

0.10

1.00

1.00

0.99

0.98

0.96

0.94

0.91

0.88

0.84

0.80

0.75

0.70

0.64

0.58

0.51

0.44

0.36

0.28

0.19

0.10

1.00

1.00

0.99

0.98

0.96

0.94

0.92

0.89

0.85

0.82

0.77

0.73

0.67

0.62

0.56

0.49

0.42

0.34

0.27

0.18

0.09

1.00

1.00

0.99

0.98

0.97

0.95

0.93

0.90

0.87

0.83

0.79

0.75

0.70

0.65

0.60

0.54

0.47

0.40

0.33

0.25

0.17

0.09

1.00

1.00

0.99

0.98

0.97

0.95

0.93

0.91

0.88

0.85

0.81

0.77

0.73

0.68

0.63

0.57

0.52

0.45

0.39

0.32

0.24

0.17

0.09

1.00

1.00

0.99

0.98

0.97

0.96

0.94

0.91

0.89

0.86

0.83

0.79

0.75

0.71

0.66

0.61

0.56

0.50

0.44

0.37

0.31

0.23

0.16

0.08

1.00

1.00

0.99

0.99

0.97

0.96

0.94

0.92

0.90

0.87

0.84

0.81

0.77

0.73

0.69

0.64

0.59

0.54

0.48

0.42

0.36

0.29

0.23

0.15

0.08

B.4 Stiffness Distribution Values

Actual values for stiffness distribution method 2 organized in a table.

Tab

le B

-2:

Sti

ffn

ess

Dis

trib

uti

on V

alu

es u

sed

for

Met

hod

2

35

APPENDIX B. TCL CODE

The next three sections contain the Tcl code written or modified by Reed Crosby to

perform the analysis and record the results. The code is broken into three parts; the main code,

the input file, and the procedures. This may be reused as a whole program or as parts and

procedures.

B.1 Main Code

# File Name: Main.tcl # # This is the main program # # Calls the input file for values of variables # Calls the Procedures file to have access to written # procedures. # # Reads a file and creates a list of all earthquakes to # run in the program # Sets MainRun = 1 to know when to stop # # The outer loop runs through the degrees of freedom. After # each run, the DOF will be increased until the max # DOF is reached determined in the input file. # Creates a list of all the masses called MList # Resets the desired period to the minimum period # # The first inner loop increases the period after each run # until the maximum period is reached # # The next inner loop runs an analysis of each earthquake in # the list of earthquakes read from the file. # wipe clears all the previous analyses # ReadRecord reads the earthquake file and turns it into # a file that can be used in the program for analysis. # # The next few lines create a list of the periods to put # at the top of the final output file.

36

# # The model is then built with the Build_Model procedure # which returns the stiffness value so that the next # analysis will start with the previous stiffness. # This reduces run time. # # There is an option to run a stat load test. The static load # was used to verify the model was being built correctly. # # The Record_Max procedure is called. # # Three different analysis procedures had to be built for # SDOF, 2DOF and 3+DOF models. The if statements make # sure the right one is used for the model. # # The next part checks the Record_Value and modifies the value # depending on what parameter is used and stored the # value in a list. # # Once it exits the second loop, the values are averaged # in order to find the average maximum value. # This value is then stored in a list called AVGList and # each value is added to the list. # # Once it exits the first loop, the values are printed in # the output file. The first line gives the minDOF, # maxDOF, damping, parameter recorded, and what # stiffness distribution was used. # Then the values for the periods used are printed to the file # and the maximum average list is printed below it for # each DOF that is run. # # At the end of the run it puts "Done with EQ!" # wipe; source input.tcl source Procedures.tcl set infile [open "$EQ_List.OUT" r] set Earthquakes [read $infile] set MainRun 1 for {set DOFRun $minDOF} {$DOFRun <= $maxDOF} {incr DOFRun} { puts "Number of DOF's: $DOFRun" set MList [DistribMSame $M $DOFRun] set Tn $startTn for {set run 1} {$Tn < $MaxPer} {set Tn [Increase_Tn $Tn $dTn]} { set EQRun 0 foreach Earthquake $Earthquakes { wipe; set infile "Earthquakes/$Earthquake.at2" set outfile "Earthquakes/$Earthquake.dat"

37

ReadRecord $infile $outfile Dt numPts if {$EQRun == 0} { if {$TnList == "$Tn"} { } else { set TnList [split "$TnList $Tn"] } } set K [Build_Model $K $M $Tn $dampR $DOFRun $PI $DistribK] set Wn [expr sqrt($K/$M)] #Static_Load 1000 #puts "Done with static!" Record_Max $DataFile $DOFRun $Record_Value $run if {$DOFRun == 1} { Run_EarthquakeSDOF $Earthquake $dampR $Wn $g $run $Dt $numPts } if {$DOFRun == 2} { Run_Earthquake2DOF $Earthquake $dampR $damping $DOFRun $g $run $Dt $numPts } if {$DOFRun > 2} { Run_EarthquakeMDOF $Earthquake $dampR $damping $DOFRun $g $run $Dt $numPts } set sumM [sumList $MList] if {"$Record_Value" == "reaction"} { set max [expr [max_file $DataFile]/$sumM/32.2/12] if {$EQRun == 0} { set AccList $max } else { set AccList [split "$AccList $max"] } } else { if {"$Record_Value" == "accel"} { set max [expr [max_file $DataFile]/32.2/12] if {$EQRun == 0} { set AccList $max } else { set AccList [split "$AccList $max"] } } else { set max [expr [max_file $DataFile]] if {$EQRun == 0} { set AccList $max } else { set AccList [split "$AccList $max"] } } } incr EQRun #puts "EQ run $EQRun, Period run $run" }

38

set Average [avgList $AccList] if {$run == 1} { set AVGList $Average } else { set AVGList [split "$AVGList $Average"] } incr run } if {$MainRun == 1} { set reportfile [open "$OutFile.out" w] puts $reportfile "Min_DOF $minDOF Max_DOF $maxDOF Damping $dampR Output $Record_Value K_Distr $DistribK \n" puts $reportfile "$TnList" } puts $reportfile "$AVGList" incr MainRun } close $reportfile puts "Done with EQ!" wipe;

B.2 Input File

# File Name: input.tcl # # This sets variables needed to run the program that can be changed # depending on the outputs desired. # # M is the mass assigned to each Node in units of kip*s^2/in # K is the starting stiffness. The stiffness will be changed in the # program so that the desired natural period is reached for each # model. Units of Kip/in # # startTn is the starting period. put in any number >= 0 (Units of sec) # MaxPer is the maximum period desired. This should be greater than the # starting period but less than 25 in order to minimize run time. # # dTn is the basic period time step. Values between 0.01 and 0.1 are # recommended # dampR is the damping used. Values between 0 and 0.1 are recommended # # minDOF is the number of DOFs for the first run. Must be an integer >0 # maxDOF is the maximum number of DOFs. Must be an integer >= minDOF. # # TnList starts a list with all of the periods used

39

# DOFList starts a list with all of the DOFs. # # The for loop creates the DOF list from minDOF to maxDOF # # Record_Value is the output the user wants to record and should be one # of the following: # disp (displacement) - inches # vel velocity* - cm/s # accel acceleration* - g # incrDisp incremental displacement - cm # "eigen i" eigenvector for mode i - unitless # reaction nodal reaction - normalized # (base shear/total wt) # # damping is chosen between one of the following: # Rayleigh Uses Rayleigh Damping # Mass Mass Damping Only # Stiffness Stiffness Damping Only # If it doesn't match one of these, Rayleigh damping is used. # # DataFile is the name of the file for the records of each iteration # OutFile is the name of the file with all of the final results # EQ_List is the name of the file with a list of all of the earthquakes # that will be run in the program. Must be saves as an .OUT file # # DistibK determines if the stiffness is uniform or non-uniform. For # uniform put "Same" and for non-uniform put "Different" # # dt is the default time step for the earthquake files. The real value # will be pulled off of the actual ground motion file. # nPts is the default number of points in the earthquake file. # # g is the constant for gravity (32.2*12). This value is the constant # that the unscaled ground motions will be multiplied by. If using # scaled ground motions, this value might need to be changed. (in/s^2) # PI is the constant for pi (3.14...) set M 1 set K 1 set startTn 0 set MaxPer 10 set dTn 0.05 set dampR 0.05 set minDOF 1 set maxDOF 25 set TnList $startTn set DOFList $minDOF for {set i $minDOF} {$i <= $maxDOF} {incr i} { set j [expr $i+1] set DOFList [split "$DOFList $j"]

40

} set Record_Value reaction set damping Rayleigh set DataFile Max set OutFile "Report" set EQ_List EQ1 set DistribK Same set dt 0.005 set nPts 3000 set g 386.4 set PI [expr 2.0 * asin(1.0)]

B.3 Procedures

# List of Procedures # Increase_Tn # Create_Nodes # DistribMSame # DistribTest # DistribKSame # DistribKDiff # Build_Model # Record_Max # maximum_list # maximum_list # max_file # Run_EarthquakeSDOF # Run_Earthquake2DOF # Run_EarthquakeMDOF # Write_2Results # Write_Results # search_PEER # avgList # sumList # ReadSMDFile.tcl # ReadNGAFile.tcl # Read_Record.tcl # Static_Load # Procedures and explanations: # Procedure Name: Increase_Tn # # This Procedure increases the Natural Period of the system by an # amount depending on the period of the system. # # Increase_Tn receives a period and a change in period increment # It then returns the new period value. It changes the period

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# a given amount depending on the current period value. # # # Created by Reed Crosby # Sept 17, 2012 # proc Increase_Tn {Tn dTn} { if {$Tn < 1} { set Tn [expr ($Tn + $dTn)] } if {$Tn < 4} { if {$Tn >= 1} { set Tn [expr ($Tn + 2*$dTn)] } } if {$Tn < 10} { if {$Tn >= 4} { set Tn [expr ($Tn + 10*$dTn)] } } if {$Tn >= 10} { set Tn [expr ($Tn + 20*$dTn)] } return $Tn } # Procedure Name: Create_Nodes # # This code creates a simple model of masses attached by # elements. The elements are assigned a material. The first # node is fixed and there is one degree of freedom for # all of the nodes. # The procedure receives a list with the stiffness values (KList) # and another list with the mass values at each node (MList). It # Also takes a damping constant, c and the number of DOFs. It does # not return anything # # The nodes are all placed on top of eachother and zerolength # elements that are made up of uniaxial materials are used # to connect the nodes. # # Created by Reed Crosby # Sept 17, 2012 proc Create_Nodes {KList MList c DOF} { wipe; model basic -ndm 1 -ndf 1 set nodestep 0 for {set NumNodes [expr $DOF + 1]} {$nodestep <= $DOF} {incr nodestep} {

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set M [lindex $MList $nodestep] if {$nodestep < 1} { set i 1 } node $i 0 -mass $M #puts "created node $i" incr i } fix 1 1 set El_Mat 1 set el_truss 1 set st_el 1 set end_el 2 #Create the materials and elements and apply to nodes foreach K $KList { uniaxialMaterial Elastic $El_Mat $K; element zeroLength $el_truss $st_el $end_el -mat $El_Mat -dir 1 incr El_Mat incr el_truss incr st_el incr end_el } } # Procedure Name: DistribMSame # # This procedure creates list that has all the same # numbers in it that is less then the number of DOFs # The procedure receives two numbers: the first number # is the value that is assigned to each value within the # list. The secong number is the number of values in # the list. # # # Created by Reed Crosby # November 12, 2012 proc DistribMSame {Dnum numL} { set List 0 for {set i 0} {$i < $numL} {incr i} { set List [split "$List $Dnum"] } return $List } # Procedure Name: DistribTest # # This procedure creates list that was used for testing # the 3 DOF systems by making the 1st and 2nd masses # very small compared to the third mass. # The procedure receives two numbers: the first number # is the value that is assigned to each value within the # list. The second number is the number of values in

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# the list. The procedure was also used for 2 DOF tests # but has to be modified. One of the small numbers must # be deleted or made so that there is only one small mass. # It then returns the list that was created. # # # Created by Reed Crosby # November 12, 2012 proc DistribTest {Dnum numL} { set List 0 set List [split "$List 0.001"] set List [split "$List 0.001"] set List [split "$List $Dnum"] return $List } # Procedure Name: DistribKSame # # This procedure creates list that has all the same # numbers in it that is less then the number of DOFs # The procedure receives two numbers: the first number # is the value that is assigned to each value within the # list. The second number is the number of values in # the list. It then returns the list that was created. # # # Created by Reed Crosby # November 12, 2012 proc DistribKSame {Dnum numL} { set List $Dnum for {set i 1} {$i < $numL} {incr i} { set List [split "$List $Dnum"] } return $List } # Procedure Name: DistribKDiff # # This procedure creates list that has the numbers # distributed depending on the distribution values # read from the DistribK.txt file. The number of items # in the list depends on the number of DOFs. # The procedure receives two numbers: the first number # is the value that is distributed and assigned to each # value within the list. The second number is the number # of values in the list. # The file is read into a list and then the correct # line is read from the list depending on min and max. # Once the line is read from the file, the first number # received is multiplied by the each value in the list # read from the file and assignmed to a list. It then # returns the list that was created. #

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# Created by Reed Crosby # November 12, 2012 proc DistribKDiff {Dnum numL} { set min 0 for {set i [expr $numL-1]} {$i >= 0} {set i [expr $i-1]} { set min [expr $min+$i] } set max [expr $min+$numL] set infile [open "Data/DistribK.txt" r] set DList [read $infile] set run 1 set one 1 foreach k $DList { set stiff [expr $k*$Dnum] if {$run > $min} { if {$run <= $max} { if {$one == 1} { set List $stiff incr one } else { set List [split "$List $stiff"] } } } incr run } #puts "$run" #puts "K distrib is: $DList" return $List } # Procedure Name: Build_Model # # This procedure builds the model by calling other procedures # and iterating to create the desired model. # The procedure receives a trial stiffness, a mass, a damping # value, the number of DOFs, the constant pi, and the stiffness # distribution method. It then returns a number with the # last stiffness method used to be used as the next trial # stiffness. This reduces run-time of the program. # # The file calls DistibK procedures depending on what method # is desired in the input file and then calls the DitribM # procedure. It then calls the Creat_Nodes procedure and # builds a trial model. The eigenvalue comand is used to find # The natural frequency of the model. This value is modified to # find the natural period of the trial model and then compare # this to the desired natural frequency. The stiffness is # then modified until the actual natural period is within # 0.001 of the desired natural period. The program is finished # once this condition is met and the last stiffness value used # is returned # # Created by Reed Crosby # November 10, 2012

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proc Build_Model {K M Tn dampR DOF PI DistribK} { wipe; for {set ActTn 1} {$ActTn != $Tn} {set K [expr $K+$Change]} { set Wn [expr 2.0 * $PI / $ActTn] set c [expr 2.0*$M*$Wn*$dampR] if {"$DistribK" == "Same"} { set Klist [DistribKSame $K $DOF] } else { set Klist [DistribKDiff $K $DOF] } #puts "Klist is $Klist" set Mlist [DistribMSame $M $DOF] Create_Nodes $Klist $Mlist $c $DOF #puts "Created Nodes and members!" set eigenvalues [eigen -fullGenLapack $DOF] set eig1 [string range $eigenvalues 0 12] set eig2 [expr sqrt($eig1)] set ActTn [expr (2*$PI/$eig2)] set diff [expr ($ActTn-$Tn)] if {$diff < 0.001} { if {$diff > -0.001} { #puts "Actual Period is $ActTn" set ActTn $Tn } } set Change [expr ($K*$diff)] if {$Tn > 1.5} { if {$Tn < 5} { set Change [expr ($Change/6)] } if {$Tn >= 5} { set Change [expr ($Change/10)] } } #puts "Actual Period is $ActTn" #puts "Change is $Change" } #puts "Eigen Values are $eig2" #puts "Wn is $Wn" #puts "Stiffness is $Klist" #puts "M is $Mlist" #puts "Period is $ActTn" return $K } # Procedure Name: Record_Max # # This procedure will record the maximum value specified at all the nodes of a # MDOF system. Record values are: # disp (displacement) # vel velocity* # accel acceleration*

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# incrDisp incremental displacement # "eigen i" eigenvector for mode i # reaction nodal reaction# # # The input values are the name of the file to be recorded to (DataFile), # the number of DOF's (DOF),the value you want to record (see above for details), # and the run number (run) # # Created by Reed Crosby # Sept 25, 2012 proc Record_Max {DataFile DOF Record_Value run} { set maxNode [expr $DOF+1] if {$Record_Value == "reaction"} { #recorder Node -file Data/nodesD.out -time -node $maxNode -dof 1 $Record_Value; recorder EnvelopeNode -file "Data/$DataFile.out" -node 1 -dof 1 "$Record_Value"; } else { recorder EnvelopeNode -file "Data/$DataFile.out" -node $maxNode -dof 1 "$Record_Value"; } } # Procedure Name: maximum_list # # This Procedure finds the maximum number within a list. # # maximum_list receives a list (List) and returns the maximum value from that list. # # # Created by Reed Crosby # Sept 17, 2012 # proc maximum_list {List} { set max 0 foreach j $List { if [$j > $max] { set max $j } } return $max } # max_file # # This Procedure finds the maximum number within a file. # # maximum_list receives a file(File) and returns the maximum value from that File. # # # Created by Reed Crosby

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# Sept 17, 2012 # proc max_file {File} { wipe; set max 0 set infile [open "Data/$File.out" r+] for {set one [gets $infile line]} {[gets $infile line] >= 0 } {set one 0} { if {$line > $max} { set max $line } } #puts "max is $max" close $infile return $max } # Procedure Name: Run_EarthquakeSDOF # # This Procedure was taken from the OpenSees basic examples manual from the example # named Time History Analysis of a 2D Elastic Cantilever Column. The procedure runs a # ground motion. # # The procedure receives the name of the ground motion (EQ), the damping value # (dampR), the natural frequency of the model (Wn), the gravity constant (g), # the run number (run), and the values for dt and nPts. It does not return anything. # # # # Modified by Reed Crosby # Sept 17, 2012 # proc Run_EarthquakeSDOF {EQ dampR Wn g run dt nPts} { timeSeries Path 2 -dt $dt -filePath "Earthquakes/$EQ.dat" -factor $g; pattern UniformExcitation 2 1 -accel 2; # set damping based on first eigen mode set Adamp [expr 2.0*$dampR*$Wn]; #damping coefficient for the stiffness rayleigh $Adamp 0 0 0; # set damping based on natural period #puts "a0 is $Adamp" # create the analysis

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wipeAnalysis; # clear previously-define analysis parameters constraints Plain; # how it handles boundary conditions numberer Plain; # renumber dof's to minimize band-width system BandGeneral; # how to store and solve the system of equations algorithm Linear # use Linear algorithm for linear analysis integrator Newmark 0.5 0.25 ; # determine the next time step for an analysis analysis Transient; # define type of analysis: time-dependent set duration [expr $nPts*$dt] set dtAnalysis 0.005 set Nsteps [expr int($duration/$dtAnalysis)] analyze $Nsteps $dtAnalysis } # Procedure Name: Run_Earthquake2DOF # # This Procedure was taken from the OpenSees basic examples manual from the example # named Time History Analysis of a 2D Elastic Cantilever Column and modified for # a 2 DOF model. The procedure runs a given ground motion. # # The procedure receives the name of the ground motion (EQ), the damping value # (dampR), the damping method (damping), the natural frequency of the model (Wn), # the gravity constant (g),the run number (run), and the values for dt and nPts. # It does not return anything. # # It is different from the SDOF analysis because it takes two eigenmodes # into account: the first and second. It has both a stiffness and mass proportional # part to the Rayleigh Damping value. # # Modified by Reed Crosby # November 15, 2012 proc Run_Earthquake2DOF {EQ dampR damping DOF g run dt nPts} { timeSeries Path 2 -dt $dt -filePath "Earthquakes/$EQ.dat" -factor $g; pattern UniformExcitation 2 1 -accel 2; # set damping based on first and second eigen mode set eigenvalues [eigen -fullGenLapack $DOF] set j 1 foreach i $eigenvalues { if {$j == 1} { set W1 $i } if {$j == 2} { set W2 $i

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} incr j } if {"$damping" == "Mass"} { set Adamp [expr 2.0*$dampR*$W1] rayleigh $Adamp 0 0 0; #puts "a0 is $Adamp" } else { if {"$damping" == "Stiffness"} { set Bdamp [expr 2.0*$dampR/($W1)] rayleigh 0 0 $Bdamp 0; #puts "a1 is $Bdamp" } else { set Adamp [expr 2.0*$dampR*$W1*$W2/($W1+$W2)]; #damping coefficient for the stiffness set Bdamp [expr 2.0*$dampR/($W1+$W2)] rayleigh $Adamp 0 $Bdamp 0; # set damping based on first and second natural periods #puts "a0 is $Adamp and a1 is $Bdamp" } } # create the analysis wipeAnalysis; # clear previously-define analysis parameters constraints Plain; # how it handles boundary conditions numberer Plain; # renumber dof's to minimize band-width (optimization), if you want to system BandGeneral; # how to store and solve the system of equations in the analysis algorithm Linear # use Linear algorithm for linear analysis integrator Newmark 0.5 0.25 ; # determine the next time step for an analysis analysis Transient; # define type of analysis: time-dependent set duration [expr $nPts*$dt] set dtAnalysis 0.005 set Nsteps [expr int($duration/$dtAnalysis)] analyze $Nsteps $dtAnalysis } # Procedure Name: Run_EarthquakeMDOF # # This Procedure was taken from the OpenSees basic examples manual from the example # named Time History Analysis of a 2D Elastic Cantilever Column and modified for # a MDOF model. The procedure runs a given ground motion. # # The procedure receives the name of the ground motion (EQ), the damping value # (dampR), the damping method (damping), the natural frequency of the model (Wn), # the gravity constant (g),the run number (run), and the values for dt and nPts.

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# It does not return anything. # # It is different from the 2DOF analysis because it takes two eigenmodes # into account: the first and the third. It has both a stiffness and mass # proportional part to the Rayleigh Damping value. # # Modified by Reed Crosby # November 15, 2012 # proc Run_EarthquakeMDOF {EQ dampR damping DOF g run dt nPts} { timeSeries Path 2 -dt $dt -filePath "Earthquakes/$EQ.dat" -factor $g; pattern UniformExcitation 2 1 -accel 2; # set damping based on first and third eigen mode set eigenvalues [eigen -fullGenLapack $DOF] set j 1 foreach i $eigenvalues { if {$j == 1} { set W1 $i } if {$j == 3} { set W3 $i } incr j } if {"$damping" == "Mass"} { set Adamp [expr 2.0*$dampR*$W1] rayleigh $Adamp 0 0 0; #puts "a0 is $Adamp" } else { if {"$damping" == "Stiffness"} { set Bdamp [expr 2.0*$dampR/($W1)] rayleigh 0 0 $Bdamp 0; #puts "a1 is $Bdamp" } else { set Adamp [expr 2.0*$dampR*$W1*$W3/($W1+$W3)]; set Bdamp [expr 2.0*$dampR/($W1+$W3)] rayleigh $Adamp 0 $Bdamp 0; #puts "a0 is $Adamp and a1 is $Bdamp" } } # create the analysis wipeAnalysis; # clear previously-define analysis parameters constraints Plain; # how it handles boundary conditions numberer Plain; # renumber dof's to minimize band-width (optimization), system BandGeneral; # how to store and solve the system of equations algorithm Linear # use Linear algorithm for linear analysis integrator Newmark 0.5 0.25 ; # determine the next time step for an analysis

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analysis Transient; # define type of analysis: time-dependent set duration [expr $nPts*$dt] set dtAnalysis 0.005 set Nsteps [expr int($duration/$dtAnalysis)] analyze $Nsteps $dtAnalysis } # Procedure Name: Write_2Results # # This Procedure takes and prints the accelerations and periods to a file. # # Write_2Results receives a file to write to (File) and two other numbers or lists # to write to that file. This is no longer used in the main file # # Created by Reed Crosby # Sept 17, 2012 # proc Write_2Results {File Tn Accel} { set outfile [open "$File.out" w] set j 0 foreach i $Tn { set Acc [lindex $Accel $j] puts $outfile "$i $Acc" incr j } close $outfile } # Procedure Name: Write_Results # # This Procedure takes and prints the one list to a file # # Write_Results receives a file to write to (File) and a list to write to that file. # This is no longer used in the main file # # Created by Reed Crosby # Oct 8, 2012 # proc Write_Results {File List} { set outfile [open "$File.out" w] puts $outfile "$List" close $outfile } # Procedure Name: search_PEER # # This procedure searches the PEER strong motion database to find Earthquakes # that meet a certain criteria. # # search_PEER receives a file to write to (File) and returns a list of the earthquakes

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# This is no longer used in the main file # # Created by Reed Crosby # October 8, 2012 # proc search_PEER {File} { set records [searchPeerNGA -fault San -magLo 6.25 -magHi 7.0] Write_Results $File $records return $records } # Procedure Name: avgList # # This procedure calculates the average value of a list of numbers # # This procedure receives a list of numbers and finds the average # value of all of those numbers and returns that value. # # Created by Reed Crosby # November 8, 2012 # proc avgList {listname} { set total [sumList $listname] set average [expr $total/[llength $listname]] return $average } # Procedure Name: sumList # # This procedure calculates the sum of a list of numbers # # This procedure receives a list of numbers and finds the sum # of all of those numbers and returns that sum. # # Created by Reed Crosby # November 8, 2012 proc sumList {List} { set total 0 foreach i $List { set total [expr $total+$i] } return $total } # The following files are referenced and MUST be included in the folder # so that the program can run. They take the records and convert # them into the type of files that can be used in the analysis. source ReadSMDFile.tcl source ReadNGAFile.tcl source Read_Record.tcl # Procedure Name: Static_Load #

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# This procedure applies a static load to a certain node and then # can output the node reactions and displacements # # This procedure receives a number that is used for the load # and prints out the nodal reactions on the screen # # Created by Reed Crosby # November 8, 2012 proc Static_Load {Load} { timeSeries Linear 1 pattern Plain 1 1 { load 2 $Load } # ------------------------------ # Start of analysis generation # ------------------------------ system BandSPD numberer RCM constraints Plain integrator LoadControl 1.0 algorithm Linear analysis Static # ------------------------------ # Start of recorder generation # ------------------------------ # create a Recorder object for the nodal displacements at node 4 recorder Node -file example.out -time -node 2 -dof 1 disp # Create a recorder for element forces, one in global and the other local system recorder Element -file eleGlobal.out -time -ele 1 forces recorder Element -file eleLocal.out -time -ele 1 basicForces # ------------------------------ # Finally perform the analysis # ------------------------------ analyze 1 # ------------------------------ # Print Stuff to Screen # ------------------------------ puts "node 0: [nodeReaction 1]" puts "node 2: [nodeReaction 2]" print node 4 }

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