C 11.3-D BUILDING Compatibility Mode

8/10/2019 C 11.3-D BUILDING Compatibility Mode

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DYNAMIC OF STRUCTURES

CHAPTER 11

RESPONSE OF 3-D BUILDING SYSTEM

Department of Civil Engineering, University of North Sumatera

Ir. DANIEL RUMBI TERUNA, MT;IP-U


2/18

UNSYMETRIC PLAN BUILDINGS SUBJECTED TO

GROUND MOTION

One-Story, Two-Way unsymmetric System

(a)

Fig.1 One storey System: (a) Plan; (b) frame B; (c) frame B and C

b

xu2d

2d

o

yu

u

( )tug&& e

Frame B

Frame C

Frame

A

(b)

AuSAf

CB uu ,

SBSC ff ,

(c)


3/18


GROUND MOTION

Force-Displacement Relation

Let refresent the vector of statically applied external forces on the

stiffness component of the structures and the vector of resulting

displacements both defined in terms of the three DOFs. The forces and

displacement are related through

uS

f

kufor

u

u

u

kkk

kkk

kkk

f

f

f

Sy

x

yx

yyyyx

xxyxx

s

Sy

Sx

=

=

(1)

The 3 x 3 stiffness matrix of the structures can be determined by the

direct equilibrium or by direct stiffness method

k


4/18


GROUND MOTION

For this pupose the lateral stiffness of each frame is defined. The lateral

stiffness of frame relates the lateral forces and

displacement (Fig. 1.b)Au

SAf

(2)

The lateral stiffness for each frame is determined by the static

condensation procedures described in Chapter 10.

yk A

The lateral stiffness of frames

,respectively, and they relates the lateral forces and displacement

shown in Fig. 1. c

xCxB kandkareCandB

AySA ukf =

CxCSCBxBSB ukfukf ==(3 )


5/18


GROUND MOTION

0,1 ===

uuu yx

1=xu0

e

xBk

xCk

xBk

xCk

xxk

yxk

x

k

( )xBxCx

yx

xCxBxx

kkd

k

k

kkk

=

=

+=

2

0


6/18


GROUND MOTION

0,1 ===

uuu xy

yx

yyy

xy

kek

kk

k

=

=

=

0

1=yu

0

e

yk

0

yk

0

xyk

yyk

x

k

0


7/18


GROUND MOTION

1,0 ===

uuu xy

( )( )xBxCy

yy

xBxCx

kkdkek

ekk

kkdk

++=

=

=

4/

2/

22

1=u

e

yek

2/dkxB

2/dkxC

2/dkxB

xk

yk

k

2/dkxC

y

ek


8/18

The complete stiffness matrix is

Alternatifvely, the stiffness matrix of tthe structures may be formulated by

the direct stiffness method (Chopra page 345)


GROUND MOTION

(4)[ ]( )

( ) ( )

++

+

=

)(4/)(2/

0

)(2/0

22

xCxByyxBxC

yy

xBxCxCxB

kkdkeekkkd

ekk

kkdkk

k

Inertia Forces

Since the selected global DOFs are located at the center of mass, the

inertia forces on the mass componenet of the structures are

t

OI

t

yIy

t

xIx uIfumfumf &&&&&& ===


9/18

Where is the diaphragm mass distributed uniformy over the plan,

is the moment of inertia of the diaphragm about the

vertical axis passing through , and

components of the total acceleration of the center of mass.

In matrix form the inertia forces and acceleration are related through the

mass matrix:


GROUND MOTION

o12/22 dbmIO +=

m

and,,theare,, yxuuu ttyt

x &&&&&&

=

t

t

y

t

x

OI

Iy

Ix

u

u

u

I

m

m

f

f

f

&&

&&

&&

00

00

00

The total acceleration are

(5)


10/18


GROUND MOTION

+

=

)(

)(

)(

tu

tu

tu

u

u

u

u

u

u

g

gy

gx

y

x

t

t

y

t

x

&&

&&

&&

&&

&&

&&

&&

&&

&&

The equation of motion excluding damping can be written as

(6)

=

+

)(

)(

)(

0

0

00

00

00

tuI

tum

tum

u

u

u

kkk

kk

kk

u

u

u

I

m

m

gO

gy

gx

y

x

yx

yyy

xxx

y

x

O

&&

&&

&&

&&

&&

&&

(7)


11/18


GROUND MOTION

where

( ) yyyxBxCxx

xCxByyyyxCxBxx

ekkkkkdkk

kkdkekkkkkk

====

++==+=

)(2/

)(4/22

The three diffrential equations in Eq. (7) are coupled through the stiffness

matrix because the stiffness properties are not symetric about the or

axes.

x y

Thus the response of the system to the or component of ground

motion is not restricted to lateral displacement in the or direction,

but will include lateral motion in the tranverse direction or , and

torsion of the roof diaphgragm about the vertical axis

x yx y

x y


12/18


GROUND MOTION

This system is symetric about the but not about the .

For this one way unsymetric system, Eq.(7) lead to

( )

=

+

+

0

)(

)(

2/0

0

002

00

00

00

22

tum

tum

u

u

u

kdkeek

ekk

k

u

u

u

I

m

m

gy

gx

y

x

xyy

yy

x

y

x

O

&&

&&

&&

&&

&&

(8)

One-Way Unsymetric System

Consider a special case of the system for which the lateral stiffness of

frame and is identical ( )B C

)(2 tumukum tgxxxx &&&& =+

axisx axisy

Where the rotational excitation is negelected. The first of three equationscan be written as

(9)

xxCxB kkk ==


13/18


GROUND MOTION

This implies that motion in the direction occurs independent of the

motion in the direction or of torsional motion.

(10)

The Eq.(8) is SDOF equation of motion that governs the response ,

of the one story ssystem to ground motion in the direction;

do not enter into this equation

uanduy

xu

( )tuI

m

u

u

kek

ekk

u

u

I

mgy

O

y

y

yyy

O

&&

&&

&&

=

+

0

1

0

0

0

0

x

y

The second and third equations can be rewritten as

x

These equation governing are coupled through the stiffnessmatrix because the stiffness properties are not symetric about the axis

. Thus the system response to the component of ground motion is

not restricted to the lateral displacement in the direction but includes

torsion about a vertical axis

uanduyy

y

y


14/18


GROUND MOTION

( )

=

+

)(

)(

)(

2/00

00

002

00

00

00

2tuI

tum

tum

u

u

u

kd

k

k

u

u

u

I

m

m

gO

gy

gx

y

x

x

y

x

y

x

O &&

&&

&&

&&

&&

&&

(11)

One- Storey Symetric System

We next consider a special case of the system for which the lateral

stiffness of frame and is identical ( ) and frame

is located at the center of mass (i.e., ). For this system Eq. (7) lead

to

B C

0=e

A

The three equations are now uncoupled, and each is of the same formas the equation for an SDOF system. This uncoupling of equtions

implies:

xxCxB kkk ==


15/18


GROUND MOTION

Ground motion in the direction would cause only lateral

motion in the direction, can be determined by solving the first

equation

x

x

Ground motion in the direction would cause only lateral

motion in the direction, can be determined by solving the

second equation

yy

The system would experience no torsional motion unless the

base motion includes rotation about a vertical axis


16/18


GROUND MOTION

Determine the transformation matrix that relates the lateral displacement

Determine The Stiffness Matrix By The Direct Stiffness Method

xu

x

of frame to , the global of system. This 1x3

A

uanduy,

B

i

y

iu

matrix is denoted by , if the frame is oriented in the direction,xia

or by , if the frame is oriented in the directionyi

a

The lateral displacement of frame , , whereeuuu yA += uau yAA =

where ( )eayA 10=

Simirlarly, the lateral displacement of frame , ,

uduu xB )2/(=or , where . Finally, the lateraluau yBB = )2/01( daxB =

displacement of frame , , or ,Cu uduu xC )2/(+= uau xCC=

where ( )2/01 daxB =The stiffness matrix for frame with respect to global DOF is

determined from the lateral stiffness or of frame in local

cordinates from

i u

xik yik i

iu


17/18

Substituting the appropriate or and or gives the stiffness


GROUND MOTION

(12)yiyi

T

yiixixi

T

xii akakorakak ==

matrix of the three frames:

yiaxia xik yik

CBA kandkk ,,

=

=20

10

000

101

0

ee

ekek

e

k yyA

=

=

4/02/

0002/01

2/01

2/

01

2dd

dkdk

d

k xBxBB

(13)

(14)


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GROUND MOTION

(15)

Finally, the stiffness matrix of the system is:

CBA kkkk ++=

=

=

4/02/

000

2/01

2/01

2/

0

1

2dd

d

kdk

d

k xCxCC

(16)

(17)

Substituting Eqs.(13), (14), and (15) gives

[ ]( )

( ) ( )

++

+

=

)(4/)(2/

0

)(2/0

22

xCxByyxBxC

yy

xBxCxCxB

kkdkeekkkd

ekk

kkdkk

k

C 11.3-D BUILDING Compatibility Mode

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