Problem 13.34
The equations governing the motion of the system in Fig.
P9.22 due to spatially varying ground motion in the x–
(b) Compare the preceding results with the response of the
system if both supports undergo identical motion ug(t).
Comment on how the responses in the two cases differ and
Condense out the rotational DOF T
uuu ],,[ 765
Solve φmφk 2
for natural frequencies and modes:
Determine nl
:
for support mode n and ground motion at support l, where
Part a
1. Determine the total displacements, u1 and u2.
1
tttt
gg
T
g
g
kkkk
kk
kk
u1
m
/
4
EI
m
/
2
m
/
4
EI
c d e
u1
u5 u6 u7
ug1 ug2
ug1
76
2
11
0.5 0.5
() ( )
Dt Dt t
(a)
2. Determine the bending moments at a, b, c, d and e.
(b)
Substitute Eq. (a) into Eq. (b) and collect terms
6
7
() 1.2 1.2
gg
L
ut
Compute bending moments in terms of nodal DOF and
Substitute Eqs. (a) and (c) into Eqs. (d)–(h) and collect
terms
22
2
() 7.2 () 7.2 ( )
bgg
EI
Mt ut utt
2
() 2.4 () 2.4 ( )
egg
EI
Mt ut utt
L
Substituting 2
/)()( nnn tAtD
:
11
22
0.0781 ( ) 0.0781 ( )
0.0075 ( ) 0.0075 ( )
mL At At t
At At t
(j)
77
2
() 2.4 () 2.4 ( )
egg
EI
Mt ut utt
Part b
For identical support motions,
Substituting Eqs. (n), (o), and (p) into Eqs. (a) and (i)–(m)
and collecting terms:
1. Total displacements, u1 and u2.
2. Bending moments at a, b, c, d, and e.
)(1563.0)( 1tmLAtMa
Problem 13.35
governing its motion due to spatially varying ground
motion in the x-direction were formulated in Problem 9.23.
of ug(t) and Dn(t), and forces in terms of ug(t) and An(t),
where Dn(t) and An(t) are the deformation and pseudo-
(b) Compare the preceding results with the response of the
structure if all column supports undergo identical motion
Preliminaries
10 2 1 0 0 0
g
T
kb
kk
1
1111
Solve φ
2m
kfor natural frequencies and mode:
Determine nl
:
for support mode n and ground motion at support l, where
Part a
Given:
)()()( ttDtDtD nndnc
,
also,
u
x
u
k
uga(t)
k
ugb(t)
b
79
1. Determine the total displacements, u, uy, and u
.
)(
0
1
)(
0
1
ttDtD
(a)
Note that there is no response in the y-direction and no
contribution to the response from the second mode.
2. Determine the shear forces in the columns.
Relate these displacements to t
x
u, t
y
u, and t
u
:
The x– and y-components of the column drifts are:
2
)()(
2
)()()()(
)()(
2
)()()()(
ttutu
b
tuttutut
tutu
b
tututut
g
t
xg
t
cxcx
g
tt
xg
t
axax
The x and y components of the column shears are:
)(
2
)()()(
)()(
2
)()()(
2
)()(
2
)()()(
tu
b
tuktktV
t
tutu
b
tuktktV
tutu
b
tuktktV
tt
ydydy
g
tt
xcxcx
g
tt
xaxax
(b)
dy
x
u
y
u
u
80
)(
4
1
)(
4
1
2
2
4
4
33
ttDtD
4
4
)(
1
)(
1
)(
1
)(
1
)( 33 ttDtDttutuktV ggcy
)(
4
)(
4
)(
4
)(
4
)( 33 ttDtDttutuktV ggdy
)(
6
1
)(
6
1
)()(
8
3311 ttAtAttAtA
m
)(
6
1
)(
6
1
)()(
8
3311 ttAtAttAtA
m
)()(
4
)(
ttutu
k
tV ggcx
Part b
For identical support motions,
Substituting Eqs. (d) into Eqs. (a) and (c) and collecting
terms:
2. The column shear forces.
If the support motions in the x-direction are identical,
only the first mode of vibration is excited, causing slab
response of the structure; i.e., the structure undergoes
coupled lateral-torsional motion. Note however that due to
the symmetry of the structure and the direction of the
Consequently, as evidenced by the above results, there is
no contribution to the response of the structure from the
second mode.
81
Problem 13.36
For the system of Fig. P13.24 the natural vibration fre-
(a) The supports of columns a and b undergo motion ug(t)
in the x-direction and the supports of columns c and d
undergo the same motion, but tˊ seconds later. Determine
the following responses as a function of time: (i) the
(b) Compare the preceding results with the response of the
structure if all column supports undergo identical motion
ug(t). Comment on how the responses in the two cases
Solution:
1. Establish data (from Problem 9.14).
DOFs: ut
x
y
uuu
2. Set up mass and stiffness matrices.
The mass matrix (from Problem 9.14) is
6
The structural stiffness matrix (from Problem 9.14) is
006
2000 21 1 2
0100 00 0 0
and the influence matrix:
3333.01667.01667.03333.0
3. Determine the natural vibration frequencies and modes.
4. Determine n
nl
nl ML .
ut
ga ()
k
2k
uy
ux
d c
82
0.0413 0.0206 0.0206 0.0413
5. Determine the response of the nth-mode SDF system
Given:
and
6. Determine the displacement response.
( ) 0.3333 0.1667
0.1667 0.3333
0.0033
t
x
ut
0.0033
0.0033 0.0033
or
83
000.5
(i)
7. Determine shears in columns.
Define nodal displacements at the top of the columns
as shown in the accompanying figure.
Relate these displacements to x
u, y
u, and
u:
ttt
2
u
b
uu xax ttt
2
u
b
uu yay
84
The x– and y-components of deformations in columns a, b, c, and d are:
)()(
2
)()()()( ttt tutu
b
tututut gxgaxax , )(
2
)()()( ttt tu
b
tutut yayay
The -x and y-components of shear in columns a, b, c, and d are:
axax kV 2 ayay kV 2
Substituting Eqs (i) and (k) into ( l ) we obtain:
)(5454.0)(5454.0
ttutuV ggay
)(3375.0)(75.0)(045.0
)(345.0)(345.0
tDtDtD
ttutuV ggbx
)(2811.0)(2337.0
31
tDtD
(m)
)(2811.0)(2337.0
)(5373.0)(5373.0
tDtD
ttutuV ggcy
)(5454.0)(5454.0
ttutuV ggdy
or, using the fact that 2
/)()( nnn tAtD , we can
write:
)(5454.0)(5454.0
ttutuV ggay
)(345.0)(345.0
ttutuV ggbx
31
)(345.0)(345.0
ttutuV ggcx
31
85
)(69.0)(69.0
ttutuV ggdx
31
8. Identical support motions.
If all the supports undergo identical motions, the
motion at the structure is given by Eq. (13.1.15), where
n
is defined by Eq. (13.1.5) with
Then Eq.(13.1.15) gives
direction are zero:
9. Comparison
If the support motions in the x-direction are identical,
only the second mode is excited, causing floor
86
Problem 13.37
quantity. Determine the peak values of the total response.
Comment on the influence of spatial variations in the
excitation.
Solution:
1. Determine ground displacement.
Double integration of )(tug
gives the ground
2. Determine )(tDn.
The properties of the three, nth–mode SDF systems
are ( 3 ,2 ,1
n)
3. Determine displacement response.
4. Determine column shears.
5. Identical support motions.
Substituting )(
2tD from Fig. P13.37b into Eq.
6. Comparison.
Identical support motions in the x-direction
cause floor displacements and column shears in the
x-direction only. However, this symmetric plan
respectively.
Table P13.37a
t
x
u0, in. 9.941
t
y
u0, in. 0.324 0
t
bu0
2, in. 0.379 0
coupled lateral- torsional response of the structure.
87
-0.4
0.4
0 5 10 15 20 25 30
-10
8.40
-6
64.297
-6
64.359
0 5 10 15 20 25 30
-6
6
Figure P13.37a
88
-10
9.941
-10
40.324
0 5 10 15 20 25 30
-10
0.379
(b/2) ut
-16
16 13.684
2.661
-16
6.842
-16
1.214
-16
6.892
-16
1.214
-16
0 5 10 15 20 25 30
-16
2.661
90
-10
4.359
-10
10
Figure P13.37e
-10
0
4
-10
-10
4
i
utx(t), in. uty(t), in.
(b/2) ut
i
91
-16
16 13.078
-16
-16
16
6.539
-16
-16
16
-16
-16
-16
0
16