BWW  drgania własne
Theoretical background of analysis (1)
The vibration of a multidegree of freedom system is described by relation
&& &
Mx + Cx + Kx = F
(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 :
&& &
DMx + DCx + x = DF (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 obcią\
eń jednostkowych.
W przypadku budynku wysokiego są to stropy budynku
z skupionymi do nich masami ścian, nadpro\
y
i ew. złączy podatnych.
Aby wyznaczyć pierwsze trzy kolumny macierzy podatności
budynku wysokiego, nale\
y więc obliczyć przemieszczenia
wszystkich tarcz stropowych od obcią\
enia tarczy stropowej
pierwszego stropu siłami: Px=1, Py=1 i Ms=1.
Obcią\
ając tarczę drugiego stropu siłami Px=1, Py=1 i Ms=1,
wyznaczamy kolumny 4..6 macierzy podatności.
Obcią\
ając 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, posłu\
ono się metodą ciągłych
połączeń.
Zaprogramowano analityczne rozwiązania dla przemieszczeń
wszystkich stropów budynku od obcią\
eń kolejnych stropów
silami jednostkowymi.
Flexibility matrix
The following systems of differential equations have been obtained for h d" z d" H
B NG (z) + A NG (z) = 0
2 2
VG (z) = - VN NG (z) (3)
2 2 2
and for 0 d" z d" h
B ND (z) + A ND (z) = FT TK
2 2
VD (z) = VT TK - VN ND (z) (4)
2 2 2
with corresponding boundary conditions
ND (0) = 0 N2 (H) = 0
G
ND (h) = NG (h) N2 (h) = N2 (h)
D G
and
VD (0) = 0 VD (0) = 0 VG2 (H) = 0
2 2
VG (h) = VD (h) VG (h) = VD (h) VG (h) = VD (h)
2 2 2 2 2 2
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 d" 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 = diag(Mk ) (k = 1, ... ,nk ) (5)
where nk - number of storeys.
The Mk 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:
M = M = Mt + (Mu + M )hu
k1,1 k2,2 w
Mk3,1 = Mk1,3 = -SMt X - (SMu X + SMw X ) hu
Mk = Mk = -SMt - (SMu + SMw ) hu
3,2 2,3 Y Y Y
Mk3,3 = Jt + ( Ju + Jw) hu
Mk1,2 = Mk2,1 = 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.
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