BWW – drgania własne

Theoretical background of analysis (1)

The vibration of a multidegree of freedom system is described by relation

F

Kx

x

C

x

M

=

+

+

&

&

&

(1)

where:

M - mass matrix,
C - damping matrix,
K - stiffness matrix,
x - d-element vector of generalized coordinates

(d - number of dynamic degrees of freedom of the structure),

F - d-element vector of generalized excitation forces, corresponding to

generalized coordinates.

Theoretical background of analysis (2)

For a shear wall multistorey structure is more natural to determine the flexibility
matrix D then the stiffness matrix K.
The vibration of a structure is described by relation :

DF

x

x

DC

x

DM

=

+

+

&

&

&

(2)

where:

C - damping matrix,
M - mass matrix,
D - flexibility matrix,
x - d-element vector of generalized coordinates

(d - number of dynamic degrees of freedom of the structure),

F - d-element vector of generalized excitation forces, corresponding to

generalized coordinates.

Zasada wyznaczania macierzy podatności D

Elementami macierzy podatności są przemieszczenia mas

od obciążeń jednostkowych.

W przypadku budynku wysokiego są to stropy budynku

z skupionymi do nich masami ścian, nadproży

i ew. złączy podatnych.

Aby wyznaczyć pierwsze trzy kolumny macierzy podatności

budynku wysokiego, należy więc obliczyć przemieszczenia

wszystkich tarcz stropowych od obciążenia tarczy stropowej

pierwszego stropu siłami: P

x

=1, P

y

=1 i M

s

=1.

Obciążając tarczę drugiego stropu siłami P

x

=1, P

y

=1 i M

s

=1,

wyznaczamy kolumny 4..6 macierzy podatności.

Obciążając kolejne stropy siłami jednostkowymi wyznaczamy

pozostałe kolumny macierzy podatności D.

Czas i dokładność wyznaczania

macierzy podatności D

W celu szybkiego i dokładnego wyznaczenia elementów

macierzy podatności D, posłużono się metodą ciągłych

połączeń.

Zaprogramowano analityczne rozwiązania dla przemieszczeń

wszystkich stropów budynku od obciążeń kolejnych stropów

silami jednostkowymi.

Flexibility matrix

The following systems of differential equations have been obtained for h

≤

z

≤

H

B N

A N

G

G

′′

+

=

( )

( )

z

z

0

′′′

= −

V

V N

G

N

G

( )

( )

z

z

(3)

and for 0

≤

z

≤

h

B N

A N

F T

D

D

T

K

′′

+

=

( )

( )

z

z

′′′

=

−

V

V T

V N

D

T

K

N

D

( )

( )

z

z

(4)

with corresponding boundary conditions

N

0

N

0

N

N

N

N

D

G

D

G

D

G

( )

( )

( )

( )

( )

( )

0

=

′

=

=

′

= ′

H

h

h

h

h

and

V

0

V

0

V

0

V

V

V

V

V

V

D

D

G

G

D

G

D

G

D

( )

( )

(

)

( )

( )

( )

( )

( )

( )

0

0

=

′

=

′′

=

=

′

= ′

′′

= ′′

H

h

h

h

h

h

h

where

h - height from the base to a point of generalized force acting,
A, B, VN, VT, FT - matrices dependent on a structure

(see Ref 21),

TK - matrix of loads, TK = diag (1,1,1),
NG(z), ND(z) - matrices containing unknown functions of the shear force

intensity in continuous connections,

VG(z), VD(z) - matrices containing functions of the horizontal displacements

of the structure.

Capital letters G, D indicate functions corresponding to the upper (z > h) and the
lower (z

≤

h) part of the structure respectively.

Mass Matrix

Using mass properties of shear walls, connecting beam bands and flexible joints as
well as floor slabs, a quasi-diagonal mass matrix of a whole structure is created

M

M

k

=

=

diag

k

n

k

(

)

(

, ... ,

)

1

(5)

where n

k

- number of storeys.

The M

k

submatrix is a symmetrical matrix of the order three. It defines inertial

properties of k-th storey. Its elements are determined in a following way:

u

w

u

t

k

k

h

M

M

M

M

M

)

(

2

,

2

1

,

1

+

+

=

=

M

M

S

S

S

h

k

k

Mt

Mu

Mw

u

X

X

X

3 1

1 3

,

,

(

)

=

= −

−

+

M

M

S

S

S

h

k

k

Mt

Mu

Mw

u

Y

Y

Y

3 2

2 3

,

,

(

)

=

= −

−

+

M

J

J

J

h

k

t

u

w

u

3 3

,

(

)

= +

+

M

M

k

k

1 2

2 1

0

,

,

=

=

where:

Mt - a mass of floor slab,
Mu - a mass of all shear walls for a system of unitary height,
Mw - a mass of all vertical bands of connecting beams and vertical flexible joints

for a system of unitary height,

SMtx,, SMty - a mass statical moments of floor slab,
SMux, SMuy - a mass statical moments of all shear walls for a system of unitary

height,

SMwx, SMwy - a mass statical moments of all vertical bands of connecting beams

and vertical flexible joints for a system of unitary height,

Jt - a mass polar moment of inertia about the Z axis of floor slab,
Ju - a mass polar moment of inertia about the Z axis of all shear walls for a system

of unitary height,

Jw - a mass polar moment of inertia about the Z axis of all vertical bands of

connecting beams and vertical flexible joints for a system of unitary height.

przejść do pliku BWW-drga-wymu