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Solution manual Mario Paz Str Dyn
Bachelor of technology in civil engineering (National Institute of Technology
Kurukshetra)
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Solution Manual for
Structural Dynamics:
Theory and Computation
Sixth Edition
Mario Paz and Young Hoon Kim
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1.1
If the weight w is displaced by amount, y, the beam and the springs will exert a total force
on the mass of
$%&
! " # ( ) *+, '
The beam and springs act in parallel. The equivalent stiffness is:
+. "
Natural frequency:
/"0
$%&
!
" # ( ) *+,
'
-
+
2 $%&
" 0 # ( ) *+,
1
3 '
Natural period:
4"
3
'
*5
" *5'0 #
,
2 $%& ) *+'(
/
1.2
Stiffness:
$%&
$ 6 789
+. " # ( ) *+, "
) * 6 7888 " :;$88<=>?@AB
'
788(
Natural frequency:
+
:$88 6 $CD
/"0 "0
" *$BE*<FGH?IJK
1
$888
Free vibration response of undamped oscillator:
-LMN " -O KPI/M )
-Q O
I@A/M
/
1
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-Q LMN " R-O /I@A/M ) -Q O KPI/M
Displacement and velocity at M " 7IJK with the initial values
-O " 7<@AB ; -Q O " *8<@AB?IJKS
*8
I@AL*$BE T 7N " R8BCU<@AB
*$BE
-Q L7N " R7 T *$BEI@AL*$BE T 7N ) *8KPIL*$BE T 7N " **BDD<@AB?IJK
-L7N " 7 T KPIL*$BE T 7N )
1.3
The stiffness of the beam is
7*%&W $%L*&X N
7* T L$8 T 78Z N T 7[8BU $ T L$8 T 78Z N T C*BE
+"V ( )
Y"
)
" *E;E[[<=>?@AB<
'
7::(
7::(
'(
Natural frequency:
\"
7 +
7 *E;E[[ 6 $CD
0 "
0
" *B*:<K]I
E8;888
*5 1 *5
1.4
a) Infinitely rigid horizontal member
Stiffness:
7* T L$8 T 78Z N T 7[7
7*%&
+ " *# ( , "
" *7;788<=>?@AB
L7* T 7EN(
'
Natural frequency:
2
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+
*7;788 6 $CD
/"0 "0
" 7CB8E<FGH?IJK
1
*E;888
\"
/
" *BC[<K]I
*5
b) Flexible horizontal member consisting of W18X30
Compute the stiffness by moment distribution method. Displace the frame horizontally by
one inch and determine the stiffness of the frame as the sum of the shear forces in both
columns. Take advantage of the symmetry by modifying the stiffness of horizontal member
by factor 3/2.
Distribution factors
+^_ "
+^a "
:%& :%
"
T 7[7 ` <7[7 ` 8B7*::
'
'
:%& $ :% $
"
T T C8* ` 7*8$ ` 8BC[ED
' *
' *
Fixed end moments:
b^_ " b_^ "
Shear force:
+"
0.1244
D%& D T L$8 T 78Z N T 7[8BU
"
'X
L7* T 7ENX
" UE8<L+ R @AB N
C$* ) CU7
T * " 7UB7:<+@]?@AB
7C8
Natural frequency:
0.8756
950 B
-118
-832
-832
832
C
A
D
950
-59
891
\"
7 +
7 7U;7:8 6 $CD
0 "
0
" *B[:<K]I
*E;888
*5 1 *5
Note:
1. Assuming a flexible girder decreases the natural frequency by only
3
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7R
*B[:
" 8B8E " Ec
*BC[
2. Adding the weight of the girder and half of the weight of the two columns decrease the
natural frequency by:
3 " *E;888 ) E8 T 7E ) * T $$ T
\"
7 +
7 7U;7:8 6 $CD
0 "
0
" *BD[<K]I
*D;*:E
*5 1 *5
7R
1.5
*BD[
" 8B8E " *BEc
*B[:
Stiffness coefficient:
+ " *#
Natural frequency:
7*%&
7U* T %&
,"
(
L*d'N
L'N(
+
7U* T %&2
/"0 "0
1
L'N( 3
\"
1.6
7E
" *D;*:ED<=>
*
/"0
/
: $ T %&2
" 0 (
*5 5 L'N 3
+
7U* T 78e T $CD
"0
" U*BD7<FGH?IJK
1
L7*8N( T E;888
-LM " *N " -O KPI/M )
-Q O
7E
I@A/M " 8BE fghLU*BD7 T *N )
I@ALU*BD7 T *N
/
U*BD7
-LM " *N " R8B:[:<@AB<
<
4
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<
-Q LMN " R-O /I@A/M ) -Q O KPI/M " RU*BD7 T 8BE T hijLU*BD7 T *N ) 7EKPILU*BD7 T *N
-Q LM " *N " R*7B8E<@AB?IJK
-k LMN " R/X -O KPI/M R -Q O /I@A/M " R/X -LMN
-k LM " *N " :;DDE<@AB?IJK X <
1.7
l bm " & T n
T
n " Go2B GKKB
For p small, I@Ap q p
R1 T 2 T ' T I@Ap " 1 T 'X T pk
8 " pk )
p " pm KPI/M )
Where / " r t is the natural frequency.
s
2
Tp
'
W= mg
pmQ
I@A/M
/
5
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1.8
l bm " & T n
+G<I@Ap
R+ T G T MGAp T G ) 1 T 2 T ' T I@Ap " 1 T 'X T pk
For p small, I@Ap q p<<<<<<MGAp q p
1 T 'X T pk ) L+ T GX R 1 T 2 T 'Np " 8
\"
a
mg
p
L
7 + T GX R 1 T 2 T '
0
1 T 'X
*5
Note: If 1 T 2 T ' u <+ T GX , the inverted pendulum is unstable.
6
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1.13
The deflection curve may be assumed to have the shape shown in Fig. 1.13 (a), which is bend
due to a force at the top of the pole.
In this case the slope p at the tip is given as
and the displacement v as
d
p"
!'X
*%&
v"
!'(
$%&
P
y
ky
a
a
p
O
Fig. 1.13 (a)
The distance G is
p
mg
a
Ry
O
O
Fig. 1.13 (b)
G"
v *
" '
p $
+"
! $%&
" (
v
'
Rx
Fig. 1.13 (c)
and is independent of p. Go a first approximation: therefore, it may be assumed that the
system is essentially equivalent to that of Fig. 1.13 (b) in which the stiffness is given by
From Fig. 1.13 (c) taking moments about O,
7
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l bm " & T n
R+ T G T KPIp T G ) 1 T 2 T ' T I@Ap " 1 T GX T pk
For small p<<<<<<KPIp q 7<<<<<<I@Ap q p<<<<<w " Gp
Then<
1 T GX T pk ) L+ T GX R 1 T 2 T GNp " 8
\"
7 + 2
7 $%& $2
0 R "
0
R
*5 1 G *5 1'( *'
1.15
Case a)
The spring constant +x for the beam is
+x "
$%&
'(
Springs are in series, therefore the equivalent stiffness is
7
7 7
7
'(
$%& ) +'(
" )
" )
"
+. + +x + $%&
$%& 6 +
+. "
$%& 6 +
$%& ) +'(
Natural frequency:
\"
7 +.
7
$%& T + T 2
0 "
0
*5 1 *5 L$%& ) +'( N3
Case b)
The spring constant of the beam is
8
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+x "
:C%&
'(
Springs are in series, therefore the equivalent stiffness is
Natural frequency:
7
7 7
7
'(
:C%& ) +'(
" )
" )
"
+. + +x + :C T %&
:C%& T +
\"
:C%& T + T 2
7 +.
7
0 "
0
*5 1 *5 L:C%& ) + T '( N3
Case c)
Deflection of simply supported beam with load P is
! T GX T >X
v"
$%& T '
So
+"
and
\"
!
$%&
" X X
v G >
7 $%& T ' T 2
0
*5 GX > X T 3
Case d)
Stiffness of the beam, from case c)
+x "
$%&'
GX 'X
9
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Spring in series:
7
7 7
7
GX >X
$%& ) + T GX > X
" )
" )
"
+. + +x + $%& T '
$%& T + T '
Natural frequency:
\"
$%& T + T 2
7 +.
7
0 "
0
*5 1 *5 L$%& T ' ) + T GX > X N3
10
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1.16
u2
u1
Newton’s Law gives: y z " 8
1W -k W
+L-X R -W N
1W -k X
1W -k W R +L-X R -W N " 8 Eq.1
1X -k X ) +L-X R -W N " 8 Eq.2
Multiplying Eq. 1 by 1X and Eq. 2 by 1W , and subtract Eq. 1 from Eq. 2
Let w " -X R -W
1W 1X L-k W R -k X N R +L1W ) 1X NL-X R -W N " 8
wk R + #
7
7
)
,w " 8
1W 1X
Natural frequency:
\"
7
7
7
0+ # )
,
1W 1X
*5
11
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2.1
The following numerical values are given:
' " 788<@AB<<<<<<<<<<<%& " 789 <=>B 1X <<<<<<<<<<<<3 " $;888<=>
+ " *;888<=>?@AB<<<<<<<<<<<<<-O " 7B8<@AB<<<<<<<<<<<-Q O " *8<@AB?IJK<<<<<<<<<<<<<<{ " 8B7E
Stiffness:
+. "
Natural frequency:
$%&
$ T 789
)
*+
"
) * T *;888 " :;$88<=>?@AB
'(
788(
/"0
+
:;$88 T $CD
"0
" *$BE*<FGH?IJK
1
$;888
/| " /}7 R { X " *$BE*}7 R 8B7EX " *$B*E<FGH?IJK
Displacement and velocity after<<<M " 7<IJKB
-LMN " ~J •€•‚ fgh<L/| M R nN
-Q LMN " R~J •€•‚ ƒ{/ fghL/| M R nN ) /| hij<L/| M R nN„
where
~ " 0-O X )
L-Q O ) -O {/NX
L*8 ) 8B7E T *$BE*NX
X)
0
"
7
" 7B:*$<@AB
/| X
*$BE*X
12
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2.2
From problem 1.6
+
7U* T 78e T $CD
/"0 "0
" U*BD7<FGH?IJK
1
L7*8N( T E;888
Natural frequency:
/| " /}7 R { X " U*BD7}7 R 8B78X " U*B7E<FGH?IJK
-LMN " ~J •€•‚ fgh<L/| M R nN
-Q LMN " R~J •€•‚ ƒ{/ fghL/| M R nN ) /| hij<L/| M R nN„
-k LMN " ~J •€•‚ ƒ*{/ /| hijL/| M R nN ) L{ X /X R /| X Nfgh<L/| M R nN„
For initial conditions (M " 8<h…f<N
~ " 0-O X )
-O " 8BE<@AB<<<<<<<<<<<-Q O " 7E<@AB?IJK<<<<<<<<<<<<<<
L-Q O ) -O {/NX
L7E ) 8BE T 8B7 T U*BD7NX
X)
0
"
8BE
" 8BE:$E<@AB
/| X
U*B7EX
†‡j n "
Final conditions (M " *<h…f<N
-Q O ) -O {/ 7E ) 8BE T 8B7 T U*BD7
"
" 8B:*D7
/| -O
8BE T U*B7E
n " ‡ˆf†‡jL8B:*D7N " *$B8C
J •€•‚ " J •OBWTeXBZWTX " UB8$D: T 78•e
fghL/| M R nN " fgh<LU*B7E T * R *$B8[[N " R8BU:DC
hijL/| M R nN " fgh<LU*B7E T * R *$B8[[N " 8B$*7U
-LM " *N " 8BE:$E< T UB8$D: T 78•e T R8BU:DC
-Q LM " *N " R8BE:$E< T UB8$D: T 78•e ƒR8B7 T U*BD7 T 8BU:DC ) U*B7E T 8B$*7U„
" R:B8C: T 78•9 @AB?IJK
-k LM " *N " 8BE:$E< T UB8$D:
T 78•e ƒ* T 8B7 T U*BD7 T U*B7E T 8B$*7U R L8B7X T U*BD7X R U*B7EX N T 8BU:DC„
" :B7C T 78•‰ @AB?IJK X
13
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2.3
The following numerical values are given:
+ " *88<=>?@AB<
1 " 78<=> T IJK X ?@AB
KŠ‹ " *Œ+1 " *Œ*88 T 78 " CUB::$
v " =A
{"
-W " 7B8<@AB
-X " 8BUE<@AB
-W
7
" =A
" 8B8E7$
-X
8BUE
v
8B8E7$
"
" 8B88C7D
*5
*5
K " { T KŠ‹ " 8B88C7D T CUB::$<=> T IJK?@AB
2.4
Ratio between first amplitude -O and amplitude after + cycles:
-O
" +v
-•
7 -O
7
7
v " <=A
"
<=A
" 8B8U7D$
+ -• 78 8B:
=A
Damping:
{"
v
8B8U7D$
"
" 8B87:EC
*5
*5
{ " 7BEc
14
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2.5
a)
for { " 7
ŠŽ•
- " L~W ) ~X MNJ •X•‚
KŠ‹ " *1/
- " L~W ) ~X MNJ ••‚
-Q " R/L~W ) ~X MNJ ••‚ ) ~X J ••‚
At M " 8
-O " ~W <<<<<<<<<<<-Q O " R/~W ) ~X <<<<<<<<<
- " ƒ-O L7 ) /MN ) -Q O M„J ••‚
b) for { u 7
Using
- " ~W J ‘’‚ ) ~X J ‘“‚
{ " Š <<<<<<<<<<<<<<KŠ‹ " *Œ+1
at M " 8;
Š
Ž•
-Q " RJ
Solving for ~W and ~X
K X +
K
!W
” 0• – R
"R
!X
*1
*1
1
/ " r•
•
!W
" R{/ ” /— | <<<<<<<</˜| " /}{ X R 7
!X
•€•‚
- " J •€•‚ ƒ~W J •—™‚ ) ~X J ••—™‚ „
{/ƒ~W J
) ~X J ••—™‚ „ ) J •€•‚ ƒ/˜| ~W J •—™‚ R ~X /˜| J ••—™‚ „
•—™ ‚
-O " ~W ) ~X
-Q O " R{/L~W ) ~X N ) /˜| L~W R ~X N
~W "
š›
X
) ›
šQ œ€•š›
X•—™
;
~X "
š›
X
R ›
šQ œ€•š›
X•—™
-O -Q O ) {/-O
-O -Q O ) {/-O
- " J •€•‚ •# )
, LfghžL/— | MN ) hijž<L/— | MNN ) # R
, LfghžL/— | MN R hijž<L/— | MNNŸ
*/˜|
*/˜|
*
*
-Q O ) {/-O
" J •€•‚ •-O fghžL/— | MN )
hijž<L/— | MNŸ
*/˜|
15
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2.6
The following values are given:
+ " $8;888
Damping force:
a)
b)
c)
d)
=>
<<<<<<<<<<<<</ " *E<FGH?IJK
@AB
z| " K-Q
z| 7888
=> T IJK
K"
"
" 7888
-Q
7B8
@AB
+
+
$8;888
=> T IJK X
/X " <<<<<<<<<<1 " X "
"
:CB8
@AB
1
/
*EX
=> T IJK X
KŠ‹ " *Œ+1 " *}$8;888 T :CB8 " *;:88
@AB
{"
4| "
*5
/}7 R { X
K
7888
"
" 8B:7D[
KŠ‹ *:88
"
*5
*EŒ7 R 8B:7D[X
" 8B*[DE<IJKB
v " <{/4| " 8B:7D[ T *E T 8B*[D[ " *BCC87
=A
-W
" <v
-X
-W
" J " J XB99OW " 7[BC7D7
-X
16
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2.7
The peaks occur whenever -Q " 8B
-Q O " 8<<<<<<say time MO
-LMN " ~J •€•‚ fgh<L/| M R nN
-Q LMN " R~J •€•‚ ƒ{/ fghL/| M R nN ) /| hij<L/| M R nN„
8 " ƒ{/ fghL/| MO R nN ) /| hij<L/| MO R nN„
Then at time M " MO ) 4|
-Q LMO ) 4| N " R~J •€•L‚› œ¡™ N ƒ{/ fghL/| MO ) /| 4| R nN ) /| hij<L/| MO
) /| 4| R nN„
/| 4| " *5
Therefore, the bracket in -Q LMO ) 4| N to equal to zero and there is a peak at this time. So peals
occurs at *5 interval in /| M
17
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2.8
The ratio
š¢
š¢£¤
may be written as
Taken log on both sides
=A #
-¥
-¥ -¥œW -¥œX
-¥œ••W
"
T
T
¦T
-¥œ• -¥œW -¥œX -¥œ(
-¥œ•
-¥
-¥ -¥œW -¥œX
-¥œ••W
, " =A •
T
T
¦T
Ÿ
-¥œ•
-¥œW -¥œX -¥œ(
-¥œ•
-¥
-¥œW
-¥œX
-¥œ••W
" =A #
, ) =A #
, ) =A #
, ) ¦ ) =A #
,
-¥œW
-¥œX
-¥œ(
-¥œ•
=A #
-¥
, " +v
-¥œ•
18
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2.10
a) The damping force is
z| " K-Q
-Q
788
=> T IJK
K"
"
" CB$$<
z|
7*
@AB
=> T IJK X
KŠ‹ " *Œ+1 " *}$;888 T 7 " 78UBE:
@AB
K
CB$$
{"
"
" 8B8[D
KŠ‹ 78UBE:
b) The damping frequency is
/| " /}7 R { X " E[B[[}7 R 8B8[DX " E:BD7<FGH?IJK
/"0
c) v is
š
d) š’ is
“
\| "
v"
$888
FGH
" E[B[[
7
IJK
/| E:BD7
"
" CBDU<K]I
*5
*5
*5{
}7 R { X
=A #
"
*5 T 8B8[D
Œ7 R 8B8[DX
-W
, " v " 8B:C
-X
" 8B:C
-W
" J " J OB§9 " 7BD7
-X
19
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2.11
From Problem 2.10
a)
=> T IJK X
KŠ‹ " *Œ+1 " *}$;888 T 7 " 78UBE:
@AB
K
*
{"
"
" 8B87C
KŠ‹ 78UBE:
b) The damping frequency is
/| " /}7 R { X " E[B[[}7 R 8B87CX " E[B[D<FGH?IJK
/| E[B[D
\| "
"
" UB7U<K]I
*5
*5
c) v is
a) The damping force is
š
d) š’ is
“
v"
*5{
}7 R { X
=A #
"
*5 T 8B87C
Œ7 R 8B87CX
-W
, " v " 8B77$
-X
" 8B77$
-W
" J " J OBWW( " 7B7*
-X
20
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2.14
Let - " wW R wX
y1
1W wkW
y2
+-
1W wk X
K-Q
Newton’s law y z " 8
1X wk X ) K-Q ) +- " 8<<<<<<<<<<<<L%¨B 7N
1W wkW R K-Q R +- " 8<<<<<<<<<<<<L%¨B *N
Multiply eq. (1) by 1W , eq. (2) by 1X and subtract
where -k " wk W R wk X
1W 1X -k ) L1W ) 1X NK-Q ) L1W ) 1X N+- " 8
2.15
Let in Problem 2.14 divide by L1W ) 1X N and let
Then, <<b-k ) K-Q ) +- " 8
Also,
Substituting,
3.2
b"
1W 1X
L1W ) 1X N
/"0
+
b
/| " /}7 R { X
K
{"
KŠ‹
-k ) *{/-Q ) /X - " 8
21
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Stiffness:
+x "
Natural frequency:
Force frequency:
:C%& :C T $8 T 78Z T 778
"
" *[;7D8<=>?@AB
L7E T 7*N(
'(
+
*[;7D8 T $CD
/"0 "0
" 78*B:<FGH?IJK
1
7888
/
©"
Frequency ratio:
U88 T *5
" U:B*E<FGH?IJK
D8
F"
Amplitude of force:
zm " 1— Jm /
©"
Amplitude of vertical motion:
ª"
zm ?+
© U:B*E
/
"
" 8BU*8:
/ 78*B:
}L7 R F X NX ) L*{FNX
"
7
U:B*EX " *$B87<=>
$CD
*$B87?*[;7D8
}L7 R 8BU*X NX ) L* T 8B7 T 8BU*NX
" 8B88$[<@AB
Note the high dynamic amplification factor ¬ " :B$[E; the frequency ratio is close to the point
of resonance, that is F " 7B
«
-®
22
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3.3
From Problem 3.2:
zm " *$B87<=>
+ " *[;7D8<=>?@AB
{ " 8B7
F " 8BU*8:
Transmissibility
4‹ "
7 ) L*{FNX
7 ) L* T 8B7 T 8BU*NX
¯¡
0
"0
"
" :B:E
L7 R F X NX ) L*{FNX
L7 R 8BU*X NX ) L* T 8B7 T 8BU*NX
zm
Force transmitted to the supports (per support):
7
7
¯ ¡ " zm 4‹ " *$B87 T :B:E " E7B*<=>
*
*
In addition, each support carries one-half of the total motor weight of 3 " 7888<=>IB
3.4
Stiffness:
Natural frequency:
7*%&
*$ T $8 T 78Z T 7[8
+"* ( "*
" *8;UU8<=>?@AB
L7E T 7*N(
'
+
*U;UU8 T $CD
/"0 "0
" 7:B*$<FGH?IJK
1
*888 T *8
/
7*
©
F" "
" 8BC:$
/ 7:B*$
Dynamic magnification factor:
°"
Steady-state amplitude:
7
7
"
" $B:ED
X
7RF
7 R 8BC:$X
23
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ª " w±‚ ° "
3.5
{ " 8B8C
From problem 3.4:
Dynamic amplification factor:
°"
zm
E;888
°"
T $B:ED " 8BC*$<@AB
+
*8;UU8
7
}L7 R F X NX ) L*{FNX
Steady-state amplitude:
ª " w±‚ ° "
zm " E;888<=>
+ " *8;UU8<=>?@AB
F " 8BC:$
"
7
}L7 R 8BC:$X NX ) L* T 8B8C T 8BC:$NX
" $B7$*
zm
E;888
°"
T $B7$* " 8B[:D<@AB
+
*8;UU8
Note: Structural damping of 8 percent of critical damping reduced in this case. The dynamic
magnification factor from 3.456 to 3.132.
3.6
a)
b)
7 ) L*{FNX
7 ) L* T 8B7 T 8BC:$NX
0
¯ ¡ " zm 0
"
E;888
" 7E;C8$<=>
L7 R F X NX ) L*{FNX
L7 R 8BC:$X NX ) L* T 8B7 T 8BC:$NX
4‹ "
¯¡
" $B7D
zm
24
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3.7
Harmonic motion
wO " 8B7
/
© " 78 T *5 " D*BC$<FGH?IJK
+<
From equation for transmissibility:
4‹ "
7 ) L*{FNX
ª
"0
L7 R F X NX ) L*{FNX
wm
Retain negative root
3<
ª " 8B871<
©M
w±L‚N " wO hij /
{"8
wm
8B7
”L7 R F X N "
"
" 78
ª 8B87
F X " 77
F " $B$7[ "
/
D*BC$
©
"
/
r$CD T +
788
+ " U$<=>?@AB
3.8
Effective force acting on the tower:
3
8B7 T 2 T I@A/
©M
2
z.²² " R78<I@AL78 T *5 T MN
z.²² " R1wk± " R
Natural Frequency:
w±‚ "
zO " 78<+@]
zO
78
"
" 8B8:
+
$888?7*
+
$888?7*
/"0 "0
" $7B8D<FGH?IJK
1
788?$CD
/| " /}7 R { X " $8BU<FGH?IJK
25
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/
© " 78 T *5 " D*BC$<FGH?IJK
F"
© D*BC$
/
"
" *B8*$
/ $7B8D
Tower relative to foundation motion by the following equation:
³"
w±‚
}L7 R F X NX ) L*{FNX
"
8B8:
}L7 R *B8*$X NX ) L* T *B8*$ T 8B7NX
" 8B87$<@AB
3.9
Transmissibility is given by
7 ) L*{FNX
7 ) L* T *B8*$ T 8B7NX
0
4‹ " 0
"
" 8B$:
L7 R F X NX ) L*{FNX
L7 R *B8*$X NX ) L* T *B8*$ T 8B7NX
3.10
First find the equivalent spring constant, that is the force to produce a unit deflection on the
beam at the location of the motor.
p<
!<
!' X
'
bO 'X
!'(
:
p"
"
"
$%&
7*%&
$%&
v<
' (
(
!
•
!' '
:– " E!'
v"
T )
7*%& :
7U*%&
$%&
(
+"
! 7U*%& 7U* T $8 T 789
"
"
" DD;D[8<=>?@AB
v
E!'(
E T 7*8(
26
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3 $$$8
=> T IJK X
1"
"
" CBD$<
2
$CD
@AB
+
FGH
/ " 0 " C[BCU
" C$U<´!b
1
IJK
Mathematical model is the damped oscillator:
"
3˜J
/
© µ+
2
ª"
}L7 R F X NX ) L*{FNX
Amplitude of motion:
/
©W "
C88 T *5
" C$B[[<FGH?IJK
D8
/
©X "
FW "
/
©( "
7888 T *5
" 78:B[*<FGH?IJK
D8
7*88 T *5
" 7*EBDD<FGH?IJK
D8
©W
©X
©(
/
/
/
" 8BUE¶<FX "
" 7B7U¶<FX "
" 7B:$
/
/
/
w±‚ "
ªLF " 7N "
3˜J X
/
© hij/
©M
2
©X 3 — T J T F X 3 — T J T F X
3— T J T /
"
"
3
+
3
2T
2T T1
2
1
w±‚ 3 — J
E8
"
"
*{
3
$$88 T * T 8B7
" 8B8[D<@AB
ªW " 8B8D:<@AB
ªX " 8B8::D<@AB
ª( " 8B8$8*<@AB
0.08
ªL@AB N
0.06
0.04
0.02
1000
800
839
1200
´!b
27
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3.14
Let the stiffness of the floor be +B Then the resonant frequency is
And the natural frequency
\· "
+
7
0
*5 1 ) 1±
\"
7 +
0
*5 1
Dividing
Then
\
1 ) 1±
"0
\·
1
\ " \· r7 )
1±
1
28
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3.15
°"
The peak will occurs when ¸‹ " 8
¸|
H°
"
HF
/¹ is the peak frequency.
ª
7
"
w±‚ }L7 R F X NX ) L*{FNX
*F¹ º7 R F¹ X » R :{ X F¹
X
F¹ X " 7 R *{ X "
/¹
/X
/¹ X " /}7 R *{ X for { ¾
ª¹ "
And
X (?X
¼º7 R F¹ X » ) º*{F¹ » ½
w±‚
*{}7 R { X
"8
W
ŒX
The corresponding phase angel is given by the following equation.
p¹ " MGA•W
*{F¹
}7 R *{ X
*{}7 R *{ X
•W
•W
"
MGA
"
MGA
*{ X
{
7 R F¹ X
Note: These results are, of course, only valid for *{ X ¾ 7 or<{ ¾
W
ŒX
" 8B[8[.
© usually small ({ ¾ 8B7N the peak frequency, /¹
Since the amount of damping in structures /
may be considered to be the same as the natural frequency /, that is, the resonant frequency.
29
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3.16
a)
\"
7 +
7 CU;888 T $CD
0 "
0
" 7CBEC<K]I
*;E*8
*5 1 *5
b)
{"
c)
K
77*
"
" 8B8[$E
KŠ‹ *}CU;888 T *E*8?$CD
From solution of Problem 3.15
ª¹ "
w±‚
*{}7 R { X
w±‚ "
d)
zO
+
zO " *{ª¹ +}7 R { X " * T 8B8[$E T 8B$[ T CU;888}7 R 8B8[$EX " :C*E<=>
At resonance
ª
7
"
w±‚ *{
zO " *{ª+ " * T 8B8[$E T 8B$[ T CU;888 " :C:8<=>
30
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3.17
From Eq. in illustrative Example 3.1
Using eq. 3.20
At resonance F " 7S
1-k ) K-Q ) +- " 1— JO /
© X I@A/
©M
1˜J X
1˜J X
/
©
F
+
1
ª"
"
}L7 R F X NX ) L*{FNX }L7 R F X NX ) L*{FNX
ª‹ "
or
*{ª‹ "
at
Solve for {
1˜J
1
F " FW
1˜J X
1 FW
ªW "
}L7 R FW X NX ) L*{FW NX
*{ª‹ FW X " ªW }L7 R FW X NX ) L*{FW NX
:{ X ª‹ X FW § " ªWX ƒL7 R FW X NX ) L*{FW NX „
{"
Force at resonance:
1˜J
*1{
ªW L7 R FW X N
*FW }ª‹ X FW X R ªW X
z‹ " 1— J/
©‹ X " 1— J
z‹ "
+
1
z‹ " *{ª‹ +
ªW ª‹ L7 R FW X N+
FW }ª‹ X FW X R ªW X
31
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3.18
Let - " wW R wX
y1
+-
1W wkW
K-Q
y2
1W wk X
1X wk X ) K-Q ) +- " zO I@A/
©<M<<<<<<<<<<<L%¨B 7N
1W wkW R K-Q R +- " 8<<<<<<<<<<<<L%¨B *N
zO I@A/
©<M
Multiply eq. (1) by 1W , eq. (2) by 1X and subtract
1W 1X -k ) L1W ) 1X NK-Q ) L1W ) 1X N+- " 1W zO I@A/
©<M<<
divide by L1W ) 1X N and let
a)
b"
1W 1X
L1W ) 1X N
b-k ) K-Q ) +- "
Steady-state solution:
b)
1W zO
I@A/
©<M<<
1W ) 1X
1W zO
I@AL/
©<M R pN
+L1W ) 1X N
-"
}L7 R F X NX ) L*{FNX
MGA<p "
where
F"
*{F
7 R FX
©
+
/
¶ </ " 0
1
/
KŠ‹ " *Œ+1
{"
K
KŠ‹
32
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4.3
zO " E+@]I
Columns are massless and girder are rigid. Neglect
damping
+W ) +X "
7*%&
)
$%&
zLÂN
M¸ " 8BD<IJK
'W
'X (
7* T $8 T 78Z T C*BC $ T $8 T 78Z T C*B
=>
+"
)
"
E777B7
)
E$UB7
"
EDE8B*
B
L7E T 7*N(
L*8 T 7*N(
@A
Then, for M ¾ M¸ S
(a)
at<M " 8BE<IJKB
-LMN "
(
Â
+
EDE8B* T $CD
/"0 "0
" 78B::<FGH?IJK
1
*8;888
-LMN "
zO
zO I@A/M
L7 R KPI/MN )
#
R M,
/
+
+M¸
I@AL78B:: T 8BEN
E;888
E;888
º7 R KPIL78B:: T 8BEN» )
V
R 8BEY
E;DE8B*
E;DE8B* T 8BD
78B::
" R8B:8[*<@AB
(b) Maximum Horizontal Deflection using Chart Fig. 4.5
4"
\PF<
*5
*5
"
" 8BD8*<IJKB
/
78B::
M
8BD
"
" 8BUU[*
4̧ 8BD8*
M
" 8BUU[ ` L°'zN•¿À " 7BEE
4̧
Á
-•¿À
" 7BEE
-±‚
-•¿À " 7BEE T -±‚ " 7BEE T
E;888
" 7B$[<@AB
E;DE8B*
33
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4.5
Determine DLF.
First interval for M ¾ M¸ S
zLÂN " zO
7 ‚
-LMN "
à zLÂN hij /LM R ÂN HÂ
1/ O
Duhamel’s integral
where
7 ‚
Â
-LMN "
à zO hij /LM R ÂN HÂ
1/ O
M¸
‚
Â
7 ‚
fgh /LM R ÂN R Ã fgh /LM R ÂN HÂ
/
/ O
Â
7
" fgh /LM R ÂN ) X hij /LM R ÂN
/
/
à  hij /LM R ÂN H "
O
Then:
-LMN "
From Eq.1 at M " M¸
‚
zO
7
•Â
T
fgh
/LM
R
ÂN
)
hij
/LM
R
ÂNŸ
1/ X M¸
/
O
zO
hij /M
"
#M R
, <<<<<<%¨B 7¶<
+M¸
/
zO
" -±‚
+
ÄÅÆ< "
First interval for M Ç M¸ S
Derivate Eq. 1
Â
¶ - " 8¶<-Q O " 8
M¸ O
-¸ "
-LMN 7
hij /M
" #M R
,
-±‚
M¸
/
zLÂN " zO ¶ <<-O " -¸ ¶<-Q O " -Q ¸
zO
hij /M¸
zO
hij /M¸
#M¸ R
, ¶ <<<<<<<-¸ " #7 R
,<
+M¸
/
+
/M¸
34
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zO
zO
L7 R fgh /MN¶<<<<<-Q ¸ "
L7 R fgh /M¸ N
+M¸
+M¸
The governing equation for the second interval is Eq. 4.4
-Q LMN "
-Q ¸
7 ‚
hij /M )
à zLÂN hij /LM R ÂN HÂ
/
1/ O
-Q ¸
zO ‚
-LMN " -¸ fgh /M ) hij /M )
à hij /LM R ÂN HÂ
1/ O
/
-Q ¸
zO
L7 R fgh /MN<<<<<<
-LMN " -¸ fgh /M ) hij /M )
/
1/ X
-LMN " -¸ fgh /M )
Plugging values of <-¸ and -Q ¸ :
zO
hij /M¸
zO
zO
L7 R fgh /M¸ N hij /M ) L7 R fgh /MN
#7 R
, fgh /M )
+
/M¸
/+M¸
+
hij /M¸
-±‚ 7 fgh /M¸
-LMN " -±‚ #7 R
, fgh /M )
# R
, hij /M ) -±‚ L7 R fgh /MN
/M¸
/ M¸
M¸
-LMN "
°'z "
-LMN
hij /M¸
7 7 fgh /M¸
" #7 R
, fgh /M ) # R
, hij /M ) L7 R fgh /MN
/M¸
/ M¸
M¸
-±‚
7
LR hij /M¸ fgh /M ) hij /M R fgh /M¸ hij /MN
" 7)
/M¸
7
Lhij /M R hij /LM ) M¸ NN
°'z " 7 )
/M¸
35
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4.6
The model of the system is the free body diagram.
w±
1
+<
w<
+Lw R w± N
1wk <
- " w R w±
wk " -k ) wk±
where
1L-k ) wk± N ) +- " 8
1-k ) +- " R1wk±
-k ) /- " Rwk±
In this case, the acceleration is a constant, wk± " 8BE2
The solution of this equation is:
The particular solution is:
-k ) /- " R8BE2
- " ¯ fgh /M ) È hij /M ) -¹
-¹ " ~;
8 ) /X ~ " R8BE2;
~"
R8BE2
¶
/X
-¹ "
Then,
R8BE2
/X
- " ¯ fgh /M ) È hij /M R
From initial conditions at M " 8¶<<<-O " 8¶<<<-Q O " 8
-O " ¯ R
8BE2
/X
8BE2
8BE2
"
8;<<<<¯
"
/X
/X
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-Q " R/¯ hij /M ) È /<fgh /M
Finally
Find -•¿À S
-•¿À "
-Q O " <È/ " 8;<<<<È " 8
-"
8BE2
8BE2
fgh /M R X
X
/
/
8BE2
hij /M " 8
/X
M " 8; -O " 8;
5
M " ¶ <<<- " -•¿À
/
-Q "
8BE2
5 8BE2
8BE2 8BE2
2
$CD
fgh / R X " R X R X " R X " R
" R$BE:<@AB
X
/
/
/
/
/
/
78UB8E
Maximum Shear forces in columns:
Left column É•¿À " +W -•¿À " E;777B7<L$BE:N " 7C;8U$<=>
Right column É•¿À " +X -•¿À " E$UB7<L$BE:N " 7;U8C<=>
37
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4.7
Repeat problem 4.6 for 10% of critical damping.
-LMN "
{ " 8B7¶ <<</ " 78B::<FGH?IJK¶<<<<+. " EDE8<=>?IJK¶<<<<<z.²² " 78;888<=>
/| " /}7 R { X " 78B::}7 R 8B7X " 78B$U<FGH?IJK
‚
7
à zLÂN J •€•L‚•ÊN hij /| LM R ÂN HÂ
1/| O
For -•¿À " 8¶
78;888 J •€•L‚•ÊN
L{/ hij /| LM R ÂN ) /| fgh /| LM R ÂNNÌ
"
Ë
1/| L{/NX ) /| X
O
J •€•‚
7
78;888
L8 ) /| N R
ƒ{/ hij /| M ) /| fgh /| M„Î "
Í
"
L{/NX ) /| X
1/| L{/NX ) /| X
J •WBO§§‚
78B$U
78;888 T $CD
ƒ7B8:: hij 78B$UM ) 78B$U fgh 78B$UM„Î
Í
R
"
*8;888 T 78B$U 7B8::X ) 78B$UX 7B8::X ) 78B$UX
" 8B7[8$EÏ78B$U R J •WBO§§‚ ƒ7B8:: hij 78B$UM ) 78B$U fgh 78B$UM„Ð
‚
H" J •WBO§§‚ ƒ7B8:: T L7B8::<hij 78B$UM ) 78B$U fgh 78B$UMN
HM
R L78B$U T 7B8::<fgh 78B$UM R 78B$UX hij 78B$UMN„ " 8
hijL78B$UMN " 8 ¶ <<<<<78B$UM " +5¶ <<<<<<+ " 8; 7; *; Ñ
M• " 8B$8*<IJK
-•¿À " 8B7[8$Eƒ78B$U R J •OB(W‰Ò L7B8::<hij 5 ) 78B$U fgh 5N„ " 8B7[8$E T 78B$U T 7B[*U
" $B8D<@AB
Left column É•¿À " tÕ -•¿À "
WXÓÔ
WXT(OTWOÖ T9X‰
LW‰TWXNÕ
Right column É•¿À " +X -•¿À " 7;D:C<=>
$B8D " 7E;D:8<=>
38
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4.19
The stiffness of the frame is
$%L*&N 7* T $8 T 78Z T * T DUB*
+"
"
" [*8CB[E<=>?@AB
'(
7*8(
1-k ) +L- R -± N " 8
1-k ) +- " +-± " zLMN
*8;888
<=>B IJK X
1"
" E7BC7:
$CD
@AB
M
0
0.25
0.5
1.0
w±
+<
zLMN
0
7208.75
0
0
1
w<
zLMN " [*8CB[E T -± LMN
+Lw R w± N
1wk <
39
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4.20
From Problem 4.19
+ " [*8CB[E<=>?@AB
1 " E7BC7<
=> T IJK X
@AB
KŠ‹ " *Œ+1 " 7***B*7
K " {KŠ‹ " 8B7 T 7***B*7 " 7**B*$
Equation of Motion:
M
0
0.25
0.5
1.0
-Q ± "
=> T IJK
@AB
1-k ) KL-Q R -Q ± N ) +L- R -± N " 8
1-k ) K-Q ) +- " K-Q ± ) +-± " zLMN
7
" :<@AB ` ¯II-1@A2<-A@\PF1<I]JJH<P\<I-]]PFM<@A<1PM@PA
8B*E
zLMN
489.0
7697.5
489.0
489.0
40
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4.25
1"
7888
=> T IJK X
" *$$B::<
$CD
@AB
$%& 7*%& $ T $8 T 78Z T 7CU*BE 7* T $8 T 78Z T 7CU*BE
+" ( ) ( "
)
" *D;U*:<L=>?@AB N
L*8 T 7*N(
L$8 T 7*N(
'W
'W
K " {KŠ‹ " *{Œ+1 " :U8BEE<
4 " *50
/
©"
=> T IJK
@AB
+
" 8BE[*:<IJK
1
78
" 7BE[*:<FGH?IJK
*5
×4 "
4
q 8B8E<IJK
78
M•¿À " *BE<IJK
78M
zLMN " 8B7 hij #
,
*5
41
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4.26
Maximum shear forces:
$%&
$ T $8 T 78Z T 7CU*BE
ÉW•¿À " ( -•¿À "
8B$:* " :*7:<=>
L*8 T 7*N(
'W
ÉX•¿À "
The bending moments:
7*%&
7* T $8 T 78Z T 7CU*BE
"
8B$:* " :UU:<=>
L$8 T 7*N(
'(X •¿À
bW•¿À " ÉW•¿À T 'W " :*7: T *:8 " 7;877;$D8<=> T @AB
bX•¿À " ÉX•¿À T 'X " :UU: T $D8 " 7;[U[;C:8<=> T @AB
And the max. stress:
ØW•¿À "
ØX•¿À "
bW•¿À 7;877;$D8
"
" $C:*<]I@
Ù
*D$B*
bX•¿À 7;[U[;C:8
"
" DC$7<]I@
Ù
*D$B*
where Ù " *D$B*<@AB( is the section modulus.
To check the calculations using the response in terms of the maximum absolute acceleration,
we consider the differential equation for relative motion,
1wk ) K-Q ) +- " 8 Eq. (a)
where
w± " support displacement
- " w R w±
Eq. (b)
Solving Eq.(a) for the total force in the column É " +-<yields
É " R1wk R w±
and replacing values from the response obtained in Problem 3.15 at time M " 7 sec. which is
the time for maximum displacement:
gives
wk•¿À " :7B$8C<@AB?IJK X <<<<<<<<<<<<<<<-Q " R8B8$D<@AB?IJK<
É•¿À " U*7*<=>
42
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This last value checks closely with the total shear force calculated using the relative
displacements in the columns, that is,
É•¿À " ÉW•¿À ) ÉX•¿À " :*7: ) :UU: " U*8C<=>
$%&
$ T $8 T 78Z T 7CU*BE
ÉW•¿À " ( -•¿À "
$BC:E " :[;$[D<=>
L*8 T 7*N(
'W
ÉX•¿À "
The bending moments:
7*%&
7* T $8 T 78Z T 7CU*BE
"
$BC:E " ED;7:D<=>
L$8 T 7*N(
'(X •¿À
bW•¿À " ÉW•¿À T 'W " :[;$[D T *:8 " 77;$[8;*:8<=> T @AB
bX•¿À " ÉX•¿À T 'X " ED;7:D T $D8 " *8;*7*;ED8<=> T @AB
And the max. stress:
ØW•¿À "
ØX•¿À "
bW•¿À 77;$[8;*:8<
"
" :$;*88<]I@
Ù
*D$B*
bX•¿À *8;*7*;ED8
"
" [D;[UE<]I@
Ù
*D$B*
where Ù " *D$B*<@AB( is the section modulus.
43
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4.27
The deflection v produced by a force ! applied at the center of a fixed beam is given by
v"
!'(
7U*%&
Therefore, the equivalent spring constant ÚÛ for the beam is
ÚÛ "
! 7U*%& 7U* T $8 T 78Z T $:C
"
"
" 7:E;888<=>?@AB
L*8 T 7*N(
v
'(
Since the coil spring and the beam are in series, the combined spring constant ÚÛ <for the system
is:
7
7
7
7
7
"
)
"
)
Ú. ÚŠ ÚÛ 7C888 7:E888
or
Ú. " 7D;87*<=>?@AB
Problem Data:
1"
\"
K"8
$888
=> T IJK X
" [B[[*<
$CD
@AB
+ " 7D;87*<L=>?@AB N
7 +
0 " [B**:<K]I
*5 1
4 " 8B7$C
×4 "
4
q 8B87<IJK
78
44
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!•¿À " +. -•¿À
4.28
+. " 7D;87*<=>
-•¿À " 8B[7
!•¿À " +. -•¿À " 7D;87* T 8B[7 " 77;$DC<=>
The maximum moment on fixed beam resulting from a concentrated force !•¿À applied at its
center is given by,
b•¿À "
!•¿À ' 77;$DCBE T *:8
"
" $:7;8EE<=> T @AB
C
C
And the maximum dynamic stress Ø•¿À by
Ø•¿À " ”
b•¿À K
$:;78EE T E
"”
" ”:U88<]I@
&
$:C
The static moment b±‚ and the static stress ر‚ resulting from the weight of the motor are then
calculated as
7
7
b±‚ " 3' " T $888 T *:8 " U8;888<=> T @AB
C
C
b±‚ K U8;888< T E
"
" 7*U$<]I@
&
$:C
The total maximum stresses are then
and
ر‚ "
Ø‚.·± " :U88 ) 7*U$ " D7U$<]I@
ØŠm•¹‹ " :U88 ) 7*U$ " D7U$<]I@
Maximum displacement of the beam:
-^;•¿À "
Maximum stretch of the coil spring:
!•¿À 77;$DCBE
"
" 8B8[C:<@AB
ÚÛ
7:E;888
-Š;•¿À " 8B[7 R 8B8[C: " 8BD$*<@AB
Maximum force zŠ;•¿À in the coil spring:
zŠ;•¿À " ÚŠ -Š;•¿À " 7C888 T 8BD$* " 77;$[D<=>B
45
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5.1
5
7888 hij 78M<< •M Ý –
78
z"Ü
5
8<< •M u –
78
/
© " 78<K]I
$%L*&N $ T $8 T 78( T * T C*BC
+W "
"
" :B$7*E<+?@AB
'(
7*8(
:B$7*E T $CD
FGH
/"0
" 77B[C
7*
IJK
/
77B[C
"
" 7BC[E<K]I
*5
*5
7
7
4" "
" 8BE$$$<IJK
\ 7BC[E
5
M¸ "
" 8B$7:7D
78
zm
7
-±‚ " "
" 8B*$*
+ :B$7*E
\"
Using Fig. 5.3
5.2
From Problem 5.1,
M
" 8BECU `
q 7BD
-±‚
4̧
-•¿À " 7BD-±‚ " 7BDL8B*$*N " 8B$[:<@AB
-•¿À " 8B$[:<@AB
+ " :B$7*E<+?@AB
% " $8 T 78( +I@
b•¿À K É•¿À 'K $%&w•¿À 'K $%&w•¿À K $ T $8 T 78( T 8B$[: T $B7
Ø•¿À "
"
"
"
"
" [B*:D
&
&
'( &
'X
7*8X
Ø•¿À " [B*:D<+I@
46
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5.3
Mathematical model:
Effective force
E
T *88 hij 78M " *BEU8[ hij 78M
$CD
/
© " 78<FGH?IJK
/
©M¸ " 5
5
5
M¸ " "
" 8B75
/
© 78
z.²² " 1<GLMN "
+"
7*%L*&W N $%&X 7* T $8 T 78( T * T C*BC $ T $8 T 78( T * T 7[8
) ( "
)
" [B:$$C<+?@AB
'(
'
*7D(
*7D(
Using Fig. 5.3
\"
7 +
7 [B:$$C T $CD
0 "
0
" $CBE<K]I
E
*5 1 *5
7
7
4" "
" 8B*D8<IJK
\ $CBE
8B75
M
"
" 7B*
4̧ 8B*D8
M
" 7B* `
q 7BD
-±‚
4̧
-±‚ "
zm *BEU8[
"
" 8B$:C<@AB
+ [B:$$C
-•¿À " 7BD-±‚ " 7BDL8B$:CN " 8BEE[<@AB
47
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5.4
Ø•¿À "
-•¿À " 8BEE[<@AB
7*%&-•¿À
<L%ÞMJFAG=<KP=-1AN
'(
$%&-•¿À
É•¿À "
<L&AMJFAG=<KP=-1AN
'(
É•¿À "
External columns:
b•¿À K É•¿À 'K
"
&
&
7*%&w•¿À 'K 7* T $8 T 78( T 8BEE[ T $BUDE
Ø•¿À "
"
" 7[B8:8U<+I@
'( &
*7DX
Internal columns:
$%&w•¿À 'K $ T $8 T 78( T 8BEE[ T :BC[E
Ø•¿À "
"
" EB*$C8<+I@
'( &
*7DX
48
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5.5
From problem 5.1:
\ " 7BC[E<K]I
From Fig. 5.8 with { " 8B7
Ù| " 7BU<@AB
Ùß " **B:<@AB?IJK
Ù¿ " 8BDC2
5.6
From Fig. 5.9 with <
And for { " 8B7
\ " 7BE[E<K]I
Ù| " :B8 T 8B$* " 7B*C<@AB
Ùß " :C T 8B$* " 7EB$D<@AB?IJK
Ù¿ " 7BE T 8B$* " 8B:C2
5.7
/"0
From Fig. 5.8 <
5.8
+
C T $CD
"0
" *B[[C<FGH?IJK
1
:88
\"
/
*B[[C
"
" 8B::*<K]I
*5
*5
Ù| " 77B8<@AB
LzÛ N•¿À " Ú T Ù| " C T 77B8 " CCB8<+@]I
From Fig. 5.8, with \ " 8B::*<K]I< and { " 8B7
Ù| " :BE<@AB
LzÛ N•¿À " Ú T Ù| " C T :BE " $DB8<+@]I
49
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5.9
Force transmits to Foundation:
1wk ) K-Q ) +- " 8
z¡ " K-Q ) +- " 1wk
Lz¡ N•¿À " 1wk•¿À " 1Ù¿
Froom Fig. 5.8 for \ " 8B::*<K]I<<and { " 8B7:
Ù¿ " 8B77U
Lz¡ N•¿À " :88 T 8B77 " ::B8<+@]I
5.10
From Problem 5.7: Max. Force in spring LzÛ N•¿À " CCB8 kips
From Fig. 5.14 with à " *B8,
yield point:
CCB8
" ::<+@]I
*
´¡ "
4"
´a " R::<+@]I
7
7
"
" *B*D<IJKS
\ 8B::*
Ù| " DB8<@AB
´¡ ::
"
" EBE
+
C
DB8
à"
" 7B7
EBE
w‚ "
From Fig. 5.14 with à " 7B*E;<<<4 " *B*D<IJK,
Ù| " 78<@AB
à"
78
" 7BC
EBE
à"
C
" 7B:E
EBE
From Fig. 5.14 with à " 7BE;<<<4 " *B*D<IJK,
Ù| " C<@AB
50
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5.11
From Problem 5.10 4 " *B*D<IJK
From Fig. 5.15 with 4 " *B*D<IJK
From Problem 5.8
à " *B8;<<<<<Ù| " $<@AB
LzÛ N•¿À " $DB88
$D
" 7C<+@]I
*
´¡ 7C
w¬¥.Ḡ"
"
" *B*E
+
C
Ù|
$B8
à"
"
" 7B$$
w¬¥.Ḡ*B*E
From Fig. 5.15 with 4 " *B*D<IJK
´¡ "
à " 7BE8<<<<Ù| " :BE<@AB
à"
Ù|
w¬¥.á¸
"
:B8
" 7BC
*B*E
5.12
a) From Fig. 5.8 with
\"
7
7
"
" *B8<K]I;<<<<</ " 7*BEDD<FGH?IJK
4 8BE
Ù| " :B8<@AB
Ùß " E8B$<@AB?IJK
Ù¿ " 7BD$2
a) From Fig. 5.9 with
\"
7
7
"
" *B8<K]I;<<<<<{ " 8
4 8BE
Ù| " 7DB8 T 8B$ " :BC<@AB
Ùß " *88 T 8B$ " D8B8<<@AB?IJK
Ù¿ " DBE:2 T 8B$ " 7BUD2
51
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5.13
a) 4 " 8BE<IJK
From Fig. 5.12 for à " :
Ù| " 8B:C<@AB6 : " 7BU*<@AB
Ùß " DB8<@AB?IJK
Ù¿ " 8B*2
b) From Fig. 5.18
\ " *B8<K]I;<<<<<<à " :
Ù| " E T : T 8B$ " DB8<@AB
Ùß " D8 T 8B$ " 7CB8<<@AB?IJK
Ù¿ " *2 T 8B$ " 8BD2
52
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6.7
Ductility factor à is defined as the ratio of maximum displacement to the yield displacement
From Problem 6.2
Max. displacements:
Yield displacements:
Ductility ratio:
-¬ "
-•¿À " *BED<@AB
+́
à"
"
$8+
" 7BE<@AB
*8+?@AB
-•¿À *BED
"
" 7B[
-¬
7BE
53
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7.1
7*%& 7* T E T 789
"
" 78*CBC7<<<=>?@AB
L7* T 7EN(
'(W
7*%& 7* T *BE T 789
+X " ( "
" 788:BDU<<<=>?@AB
L7* T 7*N(
'X
$CD8
=> T IJK X
1W "
" 78<
$CD
@AB
7U$8
=> T IJK X
1X "
" E<
$CD
@AB
+W "
For free vibration:
Non-trivial solution:
+ ) +X R 1W /X
• W
R+X
X
â*8$$BE8 R 78/
R788:BDU
GW
R+X
8
X Ÿ •GX – " •8–
+X R 1X /
R788:BDU â " 8
788:BDU R E/X
E8/§ R *8*7:B:/X ) 78$$BD$ " 8
/WX " D8B8E;<<<<<</W " [B[E<FGH?IJK
/XX " $::B*$;<<<<<</X " 7CBEE<FGH?IJK
Set /WX " D8B8E<<<<
7:$$GWW R 788:BDUGXW " 8
GWW " 7B888
GXW " 7B:*D
Set /XX " $::B*:<<<
Normalize first modes by factor:
7:$$GWX R 788:BDUGXX " 8
GWX " 7B888
GXX " R7B:8*
X
rl 1¥ G¥W
" :B:U7
X
rl 1¥ G¥X
" :B:E$
7B888
7B888
" 8B***[;<<<<ãWX "
" 8B**:D
:B:U7
:B:E$
7B:*D
R7B:8*
ãXW "
" 8B$7[E;<<<<ãXX "
" R8B$7:C
:B:U7
:B:E$
ãWW "
8B***[ 8B**:D
ƒä„ " ¼
½
8B$7[E R8B$7:C
54
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7.2
+W " E8;888<=>?@AB
+X " :8;888<=>?@AB
+( " $8;888<=>?@AB
bW " *EBU7<
3W " 78+;
=> T IJK X
;
@A
3X " D+;
bX " 7EBE:<
U8888 R *EBU7/X
ƒÚ„ " å
R:8888
8
3( " :+
=> T IJK X
=> T IJK X
;<<<<<<<<<b( " 78B$D<
@A
@A
R:8888
[8888 R 7EBE:/X
R$8888
8
æ
R$8888
X
$8888 R 78B$D/
çÚç " LU8888 R *EBU7/X NƒL[8888 R 7EBE:/X NL$8888 R 78B$D/X N R LR$8888NLR$8888N„
) :8888ƒR:8888L$8888 R 78B$D/X N„ " 8
R:7DUBD:7/Z ) :E$:D$D8/§ R 7B*7[$ T 78§ /X ) DB8 T 78W( " 8
/WX " D$$B[C;<<<<<</W " *EB7[<FGH?IJK
/XX " $*::;<<<<<</X " EDBUD<FGH?IJK
/(X " DUUD;<<<<<</( " C$BD:<FGH?IJK
LU8888 R *EBU7/X NGWW R :8888GXW " 8
GWW " 7, GXW " 7BC$UD,
R$8888GXW ) L$8888 R 78B$D/WX NG(W " 8
G(W " *B$EE7
7B8
7B8
7B8
ƒG„ " å7BC$UD 8B7:U8 R*B*C8:æ
*B$EE7 R7B*$D* 7BD78*
èW " }L*EBU7NL7NX ) L7EBE*NL7BC$UDNX ) L78B$DNL*B$EE7NX " 77BDD8*<
èX " }L*EBU7NL7NX ) L7EBE*NL8B7:U8NX ) L78B$DNL7B*$D*NX " DB:C[E<
è( " }L*EBU7NL7NX ) L7EBE*NL*B*C8:NX ) L78B$DNL7BD78*NX " 77BED8:
ã¥é "
G¥é
è¥
8B8CEC 8B7E:7
8B8CDE
ƒä„ " å8B7E[C 8B8*$8 R8B7U[$æ
8B*8*8 R8B7U8D 8B7$U$
55
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7.3
$%L*&N 7*%& 7C T 7 T 78Z
) ( "
" DB8*C<<+?@AB
L7::N(
'(W
'W
7*%L*&N *: T 78Z
+X "
"
" 7$BCCU<<+?@AB
L7*8N(
'(X
+W "
bW "
For free vibration:
Non-trivial solution:
Set /WX " 7EB*[<<<<
* T :8
=> T IJK X
" 8B*8[$<
;
$CD
@A
+ ) +X R 1W /X
• W
R+X
â7UBU7[ R 8B*8[$/
R7$BCCU
X
bX "
$ T :8
=> T IJK X
" 8B77E:<
$CD
@A
GW
R+X
8
X Ÿ •GX – " •8–
+X R 1X /
R7$BCCU
â"8
7$BCCU R 8B7EE:/X
8B8$**7/§ R EBU[:$/X ) C$B[**U " 8
/WX " 7EB*[;<<<<<</W " $BU7<FGH?IJK
/XX " 7[8B*7;<<<<<</X " 7$B8:D<FGH?IJK
7DB[E7EGWW R 7$BCCUGXW " 8
GWW " 7B888
GXW " 7B*8D
Set /XX " 7[8B$7<<<
Normalize first modes by factor:
R7EB$D[EGWX R 7$BCCUGXX " 8
GWX " 7B888
GXX " R7B78DE
X
rl 1¥ G¥W
" 8BDEC$
X
rl 1¥ G¥X
" 8BD$8D
ƒä„ " ¼7BE7U7 7BECEC ½
7BC$*7 R7B[E:[
56
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7.5
1-Wk ) *+-W R +-X " 8
1-Xk R *+-W ) *+-X R +-( " 8
In general using matrices:
Where
1-(k R +-X ) +-( " 8
ƒb„Ï-k Ð ) ƒÚ „Ï-Ð " 8
ƒb„ " 1ƒ&„ê6ê
* R7
8 8 8
í
ð
R7 * R7
8 8 8 ï
ì
R7
R7 *
8 8 8 ï
ì
*
R7
8 8 8 ï
R7
ì
ï
ƒÚ „ " ì
ì
ï
Ñ
ì
ï
Ñ
ì
ï
ì 8 8 8<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<R7 * R7 ï
ë 8 8 8<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<8 R7 7 î
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7.6
+W ) +X
ƒÚ„ " å R+X
8
R+X
+X ) +(
R+(
8
E8888 R*8888
8
R+( æ " åR*8888 $8888 R78888æ
+(
8
R78888 78888
7E8
8
8
ƒb „ " å 8
788 8 æ
8
8
E8
/WX " EUBC*:;<<<<</XX " *D8BC:;<<<<<</(X " E7*BD[;<<<<<<
8B8$7D: 8B8E:*U
8B8E*7$
ƒä„ " å8B8D:U8 8B8*UE* R8B8[87*æ
8B8U*EU R8B8U[8$
8B::CE
/W " [B[$FGHIJ
/X " 7DB7EFGHIJ
/( " **BD:<FGH?IJK<
-W " ~W ãWW hij /W M
) ~X ãWW fgh /W M ) ~( ãWX hij /X M ) ~§ ãWX fgh /X M ) ~‰ ãW( hij /( M ) ~Z ãW( fgh /( M
-X " ~W ãXW hij /W M
) ~X ãXW fgh /W M ) ~( ãXX hij /X M ) ~§ ãXX fgh /X M ) ~‰ ãX( hij /( M ) ~Z ãX( fgh /( M
-( " ~W ã(W hij /W M
) ~X ã(W fgh /W M ) ~( ã(X hij /X M ) ~§ ã(X fgh /X M ) ~‰ ã(( hij /( M ) ~Z ã(( fgh /( M
58
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8.1
Mathematical model:
zO " E888<=>
* T 7*%& * T 7* T $8 T 78Z T *:CBD
+W "
"
" $8[88<<<=>?@AB
L7* T 7EN(
'(W
* T 7*%& * T 7* T $8 T 78Z T 78DB$
+X "
"
" ::$88<<<=>?@AB
L7* T 78N(
'(X
E*E88
=> T IJK X
1W "
" 7$D<<
$CD
@AB
*EE88
=> T IJK X
1X "
" DD<<
$CD
@AB
From illustrative example 7.1 and 7.2
Modal equations:
/W " 77BC$<FGH?IJf
/X " $*BU<FGH?IJf
ƒä„ " ¼8B8D:$[ 8B8ED[ ½
8B8C7$ R8B8U*:
¨k W ) 7:8¨W " 8B8C7$ T E888 " :8DBE
¨k X ) 78C*¨X " R8B8U*: T E888 " R:D*B8
:8DBE
¨W " ¨W±‚ L7 R fgh 77BC$MN;<<<<<<<<¨W±‚ "
" *BU8
7:8
R:D*B8
¨X " ¨X±‚ L7 R fgh $*BUMN; <<<<<<<<<<<<¨X±‚ "
" R8B:*[
78C*
Solution for constant force with zero initial conditions is:
Transforming coordinate, Ï-Ð " ƒä„Ϩ Ð
-W
*BU8L7 R fgh 77BC$MN
8B8D:$[ 8B8ED[
ñ- ò " ¼
½ó
ô
X
8B8C7$ R8B8U*: R8B:*[L7 R fgh $*BUMN
-W " 8B*78U R 8B7CD[ fgh 77BC$M R 8B8*:* fgh $*BUM
-X " 8B7*E8 R 8B7D:E fgh 77BC$M ) 8B8$U: fgh $*BUM
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8.2
Use data from Problem 8.1:
Modal equations:
!W " R1W -k ± " R7$D T 8BE T $CD " R*D*:C
!X " R1X -k ± " RDD T 8BE T $CD " R7*[$C
¨k W ) 7:8¨W " ãWW !W ) ãXW !X " R8BD:$[ T *D*:C R 8B8C7$ T 7*[$C
¨k W ) 7:8¨W " 7[U$7
¨k W ) 78C*¨W " ãWX !W ) ãXX !X " R8B8ED[ T *D*:C ) 8B8U*: T 7*[$C
¨k W ) 78C*¨W " R$77B$
7[U$7
L7 R fgh 77BC$MN " 7*CB7L7 R fgh 77BC$MN
¨W "
7:8
R$77B$
L7 R fgh $*BUMN " R8B*C[:L7 R fgh $*BUMN
¨W "
78C*
Transforming coordinates
where
Ï- Ð " ƒä„ϨÐ
-‹W " -W R -±
-‹X " -X R -±
-‹W
7*CB7L7 R fgh 77BC$M N
8B8D:$[ 8B8ED[
ô
ñ- ò " ¼
½ó
8B8C7$ R8B8U*: R8B*C[:L7 R fgh $*BUMN
‹X
-‹W " CB*$ R CB*E fgh 77BC$M R 8B87D$ fgh $*BUM
-‹X " 78B:: R 78B:7 fgh 77BC$M R 8B8*DD fgh $*BUM
60
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8.3
Stiffness matrix:
Solve Eigenproblem
$888 R7E88
8
ƒÚ„ " åR7E88 $888 R7E88æ
8
R7E88 7E88
ƒb„ " å
8B$CCD
8
8
8
8B$CCD
8 æ
8
8
8B$CCD
/W X " [D:BE*<FGH?IJf
/X X " D88*B*<FGH?IJf
/( X " 7*E$$<FGH?IJf
8BE*D7:
7B7C**
8BU:C8C
ƒä„ " å8BU:C8C 8BE*D7: R7B7C**æ
7B7C** R8BU:C8C 8BE*D7:
Modal Equations:
M
,<<<<<L=>N
8B*
M
zX " *888 #7 R
,<<<<<L=>N
8B*
M
z( " $888 #7 R
,<<<<<L=>N
8B*
zW " 7888 #7 R
or
¨k W ) /W X ¨W " ãWW zW ) ãXW zX ) ã(W z(
¨k X ) /X X ¨X " ãWX zW ) ãXX zX ) ã(X z(
¨k ( ) /( X ¨( " ãW( zW ) ãX( zX ) ã(( z(
M
,<<
8B*
M
¨k X ) D88*B*¨X " RD78 #7 R
,<<<<<
8B*
¨k W ) [D:BE*¨W " EUDU #7 R
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¨k ( ) 7*E$$¨( " 7D* #7 R
M
,<
8B*
For maximum values use spectral chart Fig. 4.5
M¸ " 8B*<IJKB
/W " Œ[D:BE* " *[BDE<FGH?IJf
/X " ŒD88*B* " [[B:D<FGH?IJf
/( " Œ7*E$$ " 777BUE<FGH?IJf
4W " 8B**[*
4X " 8B8C77
4( " 8B8ED7
M¸ ?4W " 8BCC
M¸ ?4X " *B:DD
M¸ ?4( " $BEDE
L°'zNW•¿À " 7BE
L°'zNX•¿À " 7BC
L°'zN(•¿À " 7BCE
EUDU
" [BC77
[D:B*
RD78
¨X±‚ "
" R8B787D
D88*B*
7D*
¨(±‚ "
" 8B87*U*
7*E$$
¨W±‚ "
¨W•¿À " [BC77 T 7BE " 77B[7DE
¨X•¿À " R8B7C*U
¨(•¿À " 8B8*$U
Estimate maximum response using square root sum of the squares of modal contributions:
-¥•¿À " 0lºã¥éT ¨71GÞ » <<<<<<@ " 7; *; $
é
X
-W•¿À " }L8BE*D7: T 77B[7NX ) L7B7C** T 8B7C*UNX ) L8BU:C8C T 8B8*$UNX " DB7D<@AB
-X•¿À " }L8BU:8C T 77B[7NX ) L8BE*D7: T 8B7C*UNX ) L7B7C** T 8B8*$UNX " 77B8*<@AB
-(•¿À " }L7B7C** T 77B[7NX ) L8BU:C8C T 8B7C*UNX ) L8BE*D7: T 8B8*$UNX " 7$BC:<@AB
62
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8.4
The maximum shear force É¥é at story @ corresponding to mode è is given by
From 8.3
É¥é " õ¥•¿À ºã¥é R 㥕W;é »+¥
¨W•¿À " [BC77 T 7BE " 77B[7DE
¨X•¿À " R8B7C*U
¨(•¿À " 8B8*$U
8BE*D7:
7B7C**
8BU:C8C
ƒä„ " å8BU:C8C 8BE*D7: R7B7C**æ
7B7C** R8BU:C8C 8BE*D7:
ÉXW " 77B[7DEL8BU:C8C R 8BE*D7:N7E88 " [:7EBE<=>
ÉXX " R8B7C$UL8BE*D7: R 7B7C**N7E88 " 7C8B8<=>
ÉX( " 8B8*$ULR7B7C** R 8BU:C8CN7E88 " R[DB:<=>
ÉX " }[:7EBEX ) 7C8X ) [DB:X < " [:7C<=>
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8.8
7*%& 7* T 78 T 789
"
" *8EC<<<=>?@AB
L7* T 7EN(
'(W
7*%& 7* T E T 789
=>
+X " ( "
" *88U<< B
(
L7* T 7*N
@A
'X
9
7*%& 7* T E T 78
+( " ( "
" $:[*<<<=>?@AB
L78 T 7*N(
'X
+W "
$CD8
=> T IJK X
" 78<<
$CD
@AB
7U$8
=> T IJK X
1X "
" E<<
$CD
@AB
7U$8
=> T IJK X
1( "
" E<<
$CD
@AB
1W "
(a)
(b)
(c)
-W " *B*7C hij M
-X " $BUC7 hij M
-( " :B:7U hij M
-W " :B:$D fgh M
-X " [BUD$ fgh M
-( " UB7*C fgh M
-W " :B:$D fgh M ) *B*7C hij M
-X " [BUD$ fgh M ) $BUC7 hij M
-( " UB7*C fgh M ) :B:7U hij M
64
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9.1
(a)
78 R* R7 8
8
R*
D
R$
R*
ƒÚ„ " ö
÷;<<<<<ƒb„ " ö8
R7 R$ 7* R7
8
8 R* R7 C
8
Apply Gauss-Jordan elimination to two-first rows:
8
8
8
8
8
8
$
8
8
8÷
8
*
7 R8B* R8B7 8
ö8 EBD R$B* R*÷
8 R$B* 77BU R7
8 R*
R7
C
7 R8B*
R8B7
8
8
EBD
R8BE[7:
R8B$EU7
÷
ö
8 R$B*
77BU
R7
8 R*
R*B7:*C [B*CEC
7
ö8
8
8
By Eq. 9.11
© „ " ƒ4ø„¡ ƒÚ„ƒ4ø„
Check ƒÚ
8 R8B*7U$ R8B8[7:*
7 R8BE[7: R8B$EU7 ÷
8 78B8[7E R*B7:*C
8 R*B7:*C [B*CEC
8B*7U$
8B*7U$
8B8[7:*
ƒ4ø„ " ¼
½ <<<<IP<<ƒ4„ " ö8BE[7:
7
8BE[7: 8B$E[7
8
8B8[7:*
8B$E[7 ÷
8
7
78 R* R7 8 8B*7U$
8B*7U$
7
8BE[7:
8
R*
D R$ R*÷ ö8BE[7:
©„ " ¼
ƒÚ
½ö
7
8B8[7:* 8B$E[7 8 7 R7 R$ 7* R7
8 R* R7 C
8
©„ " ¼
Check ƒb
(b)
©„ " ¼ 78B8[7E R*B7:*E½
ƒÚ
R*B7:*C [B*CEC
$ 8
½
8 *
Set determinant equal to zero:
8B8[7:*
8B$E[7 ÷
8
7
©Rb
© /X „ÏGÐ " Ï8Ð<<<<<<L%¨B 7N
ƒÚ
ù78B8[7E R $/
R*B7:*C
X
R*B7:*E ù " 8
[B*CEC R $/X
/§ R [/X ) 77B:D:E " 8
/WX " *BD7$[;<<<<<</W " 7BD7D[<FGH?IJK
/XX " :B$CD;<<<<<</X " *B8U:<FGH?IJK
65
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From Eq. (1)
For /WX " *BD7$[
Set
or
For /XX " :B$CD
Set
or
ƒ78B8[7E R $L*BD7$[N„GWW R *B7:*CGWX " 8
ã(W "
ã§W "
7
Œ$ T 7B8X ) * T 7B8:8UX
7B8:8U
Œ$ T 7B8X ) * T 7B8:8UX
" 8B:$UU
" 8B:E[U
ƒ78B8[7E R $L:B$CDN„G(X R *B7:*CG§X " 8
ã(X "
ã§X "
For the first mode:
G(W " 7B8
G§W " 7B8:8U
G(W " 7B8
G§W " R7B::8U
7
Œ$ T 7B8X ) * T 7B::8UX
R7B::8U
" 8B$[$U
" R8BE$CC
Œ$ T 7B8X ) * T 7B::8UX
ã
8B:$UU
ÏãБW " • (W Ÿ " ¼
½
ã§W
8B:E[U
ã
ÏãБX " • (X Ÿ " ¼ 8B$[$U ½
ã§X
R8BE$CC
8B*7U$
ƒ4„ " ö8BE[7:
7
8
8B8[7:*
8B$E[7 ÷
8
7
8B7*[8
8B8:7D
8B:7:C
8B8*7*
ÏãÐW " ƒ4„ÏãБW " ú
û;<<<<<<< ÏãÐX " ƒ4„ÏãБX " ú
û
8B:$UU
8B$[$U
8B:E[U
R8BE$CC
66
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7
ƒb„ " ö8
8
8
9.2
From Prblem 9.1
© „ " ƒ4ø„¡ ƒb„ƒ4ø„
Check ƒb
8
7
8
8
8B*7U$
ƒ4„ " ö8BE[7:
7
8
7
8B*7U$
7
8BE[7:
8
©„ " ¼
ƒb
½ ö8
8B8[7:* 8B$E[7 8 7 8
8
8
8
$
8
8
8÷
8
*
8
7
8
8
8
8
$
8
8B8[7:*
8B$E[7 ÷
8
7
8 8B*7U$
8÷ ö8BE[7:
8
7
*
8
R8B*7U$
© „ " ¼ $B$[*
ƒb
½
R8B*7U$
*B7$$
8B8[7:*
8B$E[7 ÷
8
7
©„ " ¼ 78B8[7E R*B7:*E½
ƒÚ
R*B7:*C [B*CEC
©Rb
© /X „ÏGÐ " Ï8Ð<<<<<L%¨B 7N
ƒÚ
Set determinant equal to zero:
X
ù 78B8[7E R $B$[*/ X
R*B7:*C R 8B*7U$/
R*B7:*E R 8B*7U$/X ù " 8
[B*CEC R *B7$$/X
/§ R DBE[D[/X ) UBD*C7 " 8
/WX " *B7UU[;<<<<<</W " 7B:C$7<FGH?IJK
/XX " :B$[[;<<<<<</X " *B8U*7<FGH?IJK
67
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From Eq. (1)
For /WX " *B7UU[
ƒ78B8[7E R $B$[*L*B7UU[N„G(W R ƒ*B7:*C ) 8B*7U$L*B7UU[N„G§W " 8
G(W " 7B8
G§W " 7B877
Set
7
" 8B:8C*
*B:E
7B877
ã§W "
" 8B:7*D
*B:E
or
ã(W "
For /WX " :B$[[
ƒ78B8[7E R $B$[*L*B7UU[N„G(W R ƒ*B7:*C ) 8B*7U$L*B7UU[N„G§W " 8
G(X " 7B8
G§X " R7BE7*D
Set
or
7
" 8B$D$8
*B[EE
7B877
ã§W "
" 8BE:U8
*B[EE
ã(W "
© „ÏGÐW " ƒ7B8 7B877„ ¼ $B$[* 8B*7U$½ ¼ 7B8 ½ " *B:E
ÏGÐW¡ ƒb
8B*7U$ *B7$$ 7B877
© „ÏGÐX " ƒ7B8 R7BE7*D„ ¼ $B$[* 8B*7U$½ ¼ 7B8 ½ " *B[EE
ÏGСX ƒb
8B*7U$ *B7$$ R7BE7*D
To normalize compute:
From Problem 9.1:
ÏGÐ " ƒ4„ÏGй
8B*7U$
ƒ4„ " ö8BE[7:
7
8
8B8[7:*
8B$E[7 ÷
8
7
ã
8B:8C*
ÏãБW " • (W Ÿ " ¼
½
ã§W
8B:7*D
ã
8B$D$8
ÏãБX " • (X Ÿ " ¼
½
ã§X
R8BE:U8
8B*7:$ 8B8[7:*
8B77[8
8B77[8
8B:8C*
8B$C8C
8B$C8C
8BE[7:
8B$E[7
ÏGÐW " ú
û¼
½ " ö8B:8C*÷ ` < ÏãÐW " ö8B:8C*÷
7 8
8B:7*D
8B:7*D
8B:7*D
8 7
8B*7:$ 8B8[7:*
8B8$CD
8B8$CD
8B$D$8
8B877$
8B877$
8BE[7:
8B$E[7
ÏGÐX " ú
û¼
½ " ö 8B$D$8 ÷ ` < ÏãÐX " ö 8B$D$8 ÷
7 8
R8BE:U8
R8BE:U8
R8BE:U8
8 7
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9.5
From Problem 9.4: Natural frequencies
From Fig. 5.10: Spectral displacements:
Ù|W " DB*[:
Ù|X " 8BD77U
ý¥ " l 1é ãé¥
Participation factors:
Also,
üW " Œ7D: " 7*BC7
üX " Œ7[:C " :7BC7
ýW " 7L8B7DU7N ) 7L8B$$U[N ) 7L8BEDU[N ) 7L8BEDU[N " *B78$<
ýX " 8BD8D
Max. displacement
-¥þÿ! " rlºý¥ Ù|é ãé¥ »
-Wþÿ! " *B*:D<@AB
-Xþÿ! " :B:U*<@AB
-(þÿ! " DB887<@AB
-§þÿ! " [BE*8<@AB
-‰þÿ! " [BE*8<@AB
X
9.6
The modal shear force É¥é at story @ due to mode è is:
Where ×ã¥é " ã¥é R 㥕W;é with ãOé " 8
And É¥ " ry¥ É¥éX
É¥é " ý¥ Ù|é ×ã¥é +¥
ÉW " }L*B78: T DB*[: T 8B7DUUNX ) L8BD8D$ T 8BD77U T 8B$[EENX T 7U:U
ÉW " :$[7<=>
ÉX " :88*<=>
É( " $$*[<=>
ɧ " *$:E<=>
ɉ " 7:78<=>
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9.7
From Problem 9.5:
From Problem 9.1 and 9.4:
üWX " *B$7$7
üXX " :B7D**
78 R* R7 8
7
R*
D
R$
R*
ƒÚ„ " ö
÷;<<<<<ƒb„ " ö8
R7 R$ 7* R7
8
8 R* R7 C
8
8
7
8
8
8
8
$
8
8
8÷
8
*
[BDCDC
R*
R7 8
R*
$BDCDC
R$ R*
÷
R7 R$
EB8D8:
R7
8 R*
R7
$B$[$D
Apply Gauss-Jordan elimination to two-first rows:
ƒÚ R b/WX „ " ö
7
ö8
8
8
8 R8B$UC8 R8B7D:$
7 R7B8*U[ R8BD$7D÷
8 7BE7*U R$B8EU$
8 $B8EU$ *B778:
8B$UC8
øøø
4
ƒ4Ó „ " • Ó Ÿ " ö7B8*U[
7
&
8
ÏGÐ " ƒ4„ÏGй
The imporved modal shapes are:
8B7D:$
8BD$7D÷
8
7
ã
ÏãБW " • (W Ÿ " ¼8B7CEC½
ã§W
8B7EU*
8B$UC8
ÏGÐW " ú7B8*U[
7
8
8B$UC8
ÏGÐX " ú7B8*U[
7
8
Normalizing factor = 0.6142
ã
ÏãБX " • (X Ÿ " ¼ 8B7EDE ½
ã§X
R8B*8*U
8B7D:$
8B7887
8B*88:
8BD$7Dû ¼8B7CEC½ " ö8B*U78÷ ` < ÏãÐ " ö8BEC::÷
W
8B7CEC
8B$[*8
8
8B7EU*
8B7EU*
8B$7C[
7
8B7D:$
R8B88CE
R8B88CE
8BD$7Dû ¼ 8B7EDE ½ " öR8B8:[8÷ ` < ÏãÐ " öR8B8:[8÷
X
8B7EDE
8B7EDE
8
R8B*8*U
R8B*8*U
R8B*8*U
7
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9.9
From Problem 9.4
a)
Modal response:
Praticipation factors:
üWX " 7D:BED
üXX " 7:[$
8B7DU[
8B$:DD
8B$$U:
ÏãÐW " "
#;<<<<<<ÏãÐX " " 8BDU$* #
8BE*CD
R8B7$D8
8BD*$7
R8BEE8D
ý¥ " l 1é ãé¥
Also,
ýW " Rƒ7L8B7DU[N ) 7L8B$$U:N ) 7L8B:$:8N ) 7L8BE*EDN ) 7L8BD$$7N„ " R*B8U:E<
Spectral response
4W "
ýX " 8BD$7C
*5
*5
" 8BE88;<<<4X "
" 8B7D$[
/W
/X
M¸
8B*E
M¸ 8B*E
"
" 8B7*E;<<<<< "
" 7BE*[
4X 8B7D$[
4W 8BE8
õX•¿À
õW•¿À
" 8B$U;<<<<<
" 7B[
õX±‚
õW±‚
!O ýW :88 T *B8U:C
"
" EB8U
7D:BED
üWX
!O ýW :88 T 8BD$7C
õX±‚ " X "
" 8B7[
7:[$B:
üW
õW±‚ "
b)
õW•¿À " 8B$U T EB8U " 7BUU
õX•¿À " 7B[ T 8B7[ " 8B*U
-¥é " max. displacement at floor “@" and “è” mode relative to base displacement
-¥é " 0lºã¥é õé•¿À »
é
X
-W " }L8B7DU[ T 7BUUNX ) L8B$:DD T 8B*UNX " 8B$E*$<@AB
-X " }L8B$$U: T 7BUUNX ) L8BDU$* T 8B*UNX " 8B[7[*<@AB
-( " }L8B:$:8 T 7BUUNX ) L8B*[CD T 8B*UNX " 8BCD[:<@AB
-§ " }L8BE*C8 T 7BUUNX ) L8B7$D8 T 8B*UNX " 7B8E*D<@AB
-‰ " }L8BD*$7 T 7BUUNX ) L8BEE8D T 8B*UNX " 7B*E8*<@AB
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z¥é " $‡%B<ij…ˆ†i‡&<'gˆf…<‡†<'&ggˆ<@<ij<$g(…<èS
c)
z¥é " 1¥ ã¥é /éX õé•¿À
=> T IJK X
1¥ " 7
@AB
zWW " 7 T 8B7DU[ T 7D:BED T 7BUU " EEBE[
zXW " 7 T 8B$$U: T 7D:BED T 7BUU " 777B7:
z(W " 7:*B7*
z§W " 7[$B78
z‰W " *8:B8E
zWX " 7 T 8B$:DD T 7:[$B: T 8B*U " 7:CB78
zXX " *DUB7U
z(X " 77UB8:
z§X " RECB77
z‰X " R*$EB*D
É¥é " )‡%B<hž…‡ˆ<'gˆf…<‡†<h†gˆ*<@<ij<$g(…<è
ê
É¥é " l z•é
•+¥
ɉW " *8:B8E
ɧW " $[[B7E
É(W " E7UB*[
ÉXW " D$8B:7
ÉWW " DCEBUC
ɉX " R*$EB*D
ɧX " R*U$B$[
É(X " R7[:B$$
ÉXX " U[BC$
ÉWX " *:EBU$
É¥ " )‡%B<hž…‡ˆ<'gˆf…<‡†<h†gˆ*<@
*
ÉW " [*CB[$<=>;
É¥ " ,l É*@è
@"7
ÉX " D$[BUD<=>;<<<<<<É( " E:[BE8<=>;<<<<ɧ " :[[BC7=>;<<<<<<ɉ " $77B:*<=>
72
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10.2
1§ " 8B87 T 788 " 7< V=>
1‰ " 7
10.8
8
í8
© „ " ìì8
ƒb
ì8
ë8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
8
h…f X
Y
@AB
8
8ðï
8ï
8ï
7î
From Eq. 10.45
ì P1 ü
é 36 3L -36 3L ù ìd1 ü
ïP ï
ï
ê
2
2 úï
ï 2 ï N ê 3L 4 L -3L - L ú ïd 2 ï
í ý=
í ý
ï P3 ï 30 L ê -36 -3L 36 -3L ú ïd 3 ï
ê
2
2 ú
ë 3L - L -3L 4 L û îïd 4 þï
îï P4 þï
System geometric matrix:
$D
$88
R$D
$88
78888
$88
:8888
R$88
R78888
©- „ "
ƒÚ
<ö
÷
R$88
$D
R$88
$8 T 788 R$D
$88 R78888 R$88 :8888
C8888 R78888 8 <<<<<<<<8 <<<R$88
í
R78888 $88 8 ðï
78 ìR78888 C8888
©. „ "
ƒÚ
8
R78888
8
<<<8
$88 ï
$ ìì 8
$88
8
<<<[* R$Dï
<<<
ë R$88
8
$88
R$D <<[* î
73
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11.8
$<
7<
:<
Þ<
78\LMN<
7<
!W <
*<
w<
*<
For member 1
!W <
!‰ <
<
!§
<
!W
!X <
<
!‰
[<
B[8
8
N
LM
\
78
78\LMN8B[8[<
<
!X
w<
<
!(
<
!W
!LMN'X
" C$$B$!LMN
7*
!‰ " R
11.9
!( <
!§ <
Þ<
*<
t
t
Þ X
!LMN'X
!( " Ã R!LMN/X LÞNHÞ< " R!LMN Ã Þ •7 R – HÞ< " R
" RD$$B$!LMN
'
7*
O
O
!Z "
For member 2
7<
!LMN'X
" RE8!LMN
*
!W " !X " !§ " 8
!§ " !W " R$BE$$\LMN
!X " !‰ " $BE$$\LMN
Þ X
!( " $BE$$\LMN âÞ •7 R – â t
' À+
X
'
!( " $BE$$\LMN " ::B7C\LMN<
C
'
!Z " R$BE$$\LMN " R::B7C\LMN
C
zW " C$$B$!LMN R ::B7C\LMN
zX " RC$$B$!LMN
z( " RE\LMN
z§ " RE8!LMN
74
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12.10
!W <
!X <
õ<
!( <
w<
!LMN " !O \LMN<
w<
'<
Equivalent forces = fixed-end reaction with opposite sign.
12.11
!W " 8
7
!X " R !O \LMN'X
7*
!O \LMN'
!( "
*
!§ " 8
7
!‰ "
! \LMN'X
7* O
!O \LMN'
!Z "
*
!‰ <
!Z <
!§ <
Þ<
!W " 8
!X " R
!( "
zLMN>X
L$G ) >N
'X
!§ " 8
!‰ "
!Z "
zLMNG>X
'X
zLMNGX >
'X
zLMNGX
L$G ) >N
'X
75
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i
14.2
Using results from Problem 14.1
Modal shapes:
1st mode
Set
or
2nd mode
Set
or
çƒÚ„ R ƒb„/X ç " 8
ù7ED$ R [B*/
U8*
X
U8*
ù"8
:DCC R [B*/X
E7BC:/§ R :E;88[/X ) D;E7$;[:8 " 8
/WX " 7C$BE;<<<<<<\W " *B7D<K]I
/XX " DC:B:;<<<<<<\X " :B7D<K]I
ƒ7ED$ R [B*L7C$BEN„GWW R U8*GXW " 8
GWW " 7B8
GXW " R8B*DC
7
" 8B$EUUD
*B[[C
7B877
ãXW " R
" R8B8UD:[
*B[[C
ãWW "
ƒ7ED$ R [B*LDC:B:N„GWX R U8*GXX " 8
GWX " 7B8
GXX " $B[$
7
" 8B8UD:U
78B$D*[
$B[$
ãXX "
" 8B$EUU:
78B$D*[
ãWX "
76
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14.5
From Problem 14.2
/WX " 7C$BE;<<<<<<\W " *B7D<K]I
/XX " DC:B:;<<<<<<\X " :B7D<K]I
Natural frequencies:
4W "
ãWW " 8B$EUUD
ãXW " R8B8UD:[
ãWX " 8B8UD:U
ãXX " 8B$EUU:
7
" 8B:D$<IJK;<<<<4X " 8B*:8;<<<<M¸ " 8B7IJK
*B7D
M¸
õ7$‡%
" 8B*7D;<<<<°' zWþÿ! "
" 7B*
4¸
õW±‚¿‚
M¸
õ*$‡%
" 8B:7[;<<<<°' zXþÿ! "
" 7BU
4¸
õX±‚¿‚
¨k W ) /W X ¨W " ãWW zW ) ãXW zX
¨k X ) /X X ¨X " ãWX zW ) ãXX zX
or
¨k W ) 7C$B:¨W " 78L8B$EUUCN " $BD;<<<<M Ý 8B7<IJK<<
)
¨k X D88*B*¨X " 78L8B8UD:DN " 8BUD:D;<<<<M Ý 8B7<IJK<<<<<
õW;±‚ "
Use SSS rule
zO
$BD
zO 8BUD:D
"
" 8B7UD;<<<<<õX;±‚ "
"
" 8B887:
+
7CB$:
+
DC:BD
õW;•¿À " 7B*L8B7UDN " 8B*$E;<<< õX;þÿ! " 7BUL8B887:N " 8B88*D
wW;•¿À " rLõW;þÿ! ã77 NX )LõX;þÿ! ã7* NX < " rL8B*$E T 8B$DNX )L8B88*D T 8B8UD:DNX < " 8B8C:D<@AB
wX;•¿À " rLõW;þÿ! ã*7 NX )LõX;þÿ! ã** NX < " rLR8B*$E T 8B8UD:DNX )L8B88*D T 8B$DNX < " 8B8**[<@AB
77
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17.1
%& " $BE T 78e <=> T @ABX
=>
3 " 7E8 (
\M
%&
/· " AX 5 X 0 §
1
©'
7E8
=>
" 8B8CDC (
7*(
@AB
8B8CDC T *: T 78
=> T h…f X
1
©"
" 8B8E:
$CD
@AB(
3"
/W " 7$B:D$C;<<<<<<\7 "
/W
" *B7:<K]I
*5
/( " 7*7B7[:*;<<<<<<\$ "
/(
" 7UB*C<K]I
*5
/X " E$BCEE*;<<<<<<\* "
17.2
/X
" CBE[<K]I
*5
From Table 17.3
%&
/· " ~· 0 §
1
©'
From Problem 17.1
~W " **B$[$$;<<<<~X " D7BDC*C;<<<<<<~( " 7*8BU8$:
/· " ~·
7$B:D$C
" 7B$D:*<~·
5X
/W " $8BE*;<<<<<\W " :BCE<K]I
/X " C:B7$;<<<<<\X " 7$B$U<K]I
/( " 7D:BU$;<<<<<\( " *DB*E<K]I
78
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lOMoARcPSD|47315249
17.3
From Table 17.5
%&
/· " ~· 0 §
1
©'
7$B:D$C
/· " ~·
" 7B$D:*<~·
5X
~W " 7EB:7CC;<<<<~X " :UBUD:C;<<<<<<~( " 7:8B*:[[
From Problem 17.1
/· " ~·
7$B:D$C
" 7B$D:*<~·
5X
/W " *7B8*;<<<<<\W " $B$E<K]I
/X " DCB7D;<<<<<\X " 78BCE<K]I
/( " 7:*B*7;<<<<<\( " **BD$<K]I
17.4
wLÞ; MN "
Load
Deflection at centers
For max. deflection
w•¿À Š.·‚.‹ "
*!O
A5Þ
A5ÞW
7
L7 R fgh /· MN hij
l X hij
1
'
'
/·
©'
!O " R*888;<<<<GM<ÞW " UL7*N " 78C<@AB
Þ"
L$DNL7*N
" *7D<@AB
*
L7 R fgh /· MN " *;<<<<<1
© " 8B8E:<
=> T h…f X
;<from problem 17.1
@AB(
*
7
R*L*888N
A5L78CN
A5
A5
l X hij
hij
" R$:*BU$ l X hijL8B[CE:AN hij
/·
/·
8B8E:L$DNL7*N
L$D T 7*N
*
*
First 3 modes: /W " 7$B:D<FGH?IJK<;<<<<<From Problem 17.1<<<<
w•¿À Š.·‚.‹ " R$:*BU$ •
7
8B[8[
7
L8N )
LR7N Ñ B B Ÿ
8B[8[ )
X
X
E$BCE
7*7B7[X
7$B:D
" R$:*BU$ƒ8B88$U R 8B8888:C„
" R7B$*7<@AB
79
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lOMoARcPSD|47315249
17.5
1
© " 8B$
=> T h…f X
@AB(
%& " 78Z <=> T @ABX
Modal differential equation:
where
' " 7E8<@AB<
b· õk· LMN ) /·X b· õ·L‚N " z· LMN
t
b· " Ã 1
© ã·X LÞNHÞ
t
O
z· " Ã ã·X LÞN!LÞ; MN HÞ
O
or
õk· ) /· X õ· "
&· "
Modal response:
õ· LMN "
Total response:
zLMN !O 0 ã· LÞNHÞ
" t
b·
© ã·X LÞNHÞ
0 1
O
0 ã· LÞNHÞ
© ã·X LÞNHÞ
0O 1
t
!O &·
L7 R fgh /· MN
/· X
A5Þ
<<<$g(‡&<hž‡1…
'
·
!O &·
A5Þ
" l X L7 R fgh /· MN hij
<<<<<<<<<<
/·
'
-LM; ÞN " l ä· LÞNõ· LMN;<<<<<<<<<<ä· " hij
·
First mode
78D
*0
"
5
" 8BC
© ':
1
8B$<L7E8N:
/W " ~7 0
%&
\W "
wW LM; ÞN "
/7
*5
" 8B7$<K]I
&· "
:
5
:!O
5Þ
Þ
L7 R fgh 8B7$MN hij
" [EB$:!O L7 R fgh 8B7$MN hij
X
5<8B7$
7E8
:[B[E
80
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17.6
1
© " 8B$
=> T h…f X
@AB(
%& " 78Z <=> T @ABX
/· " ~A 0
78D
" ~A 0
" ~A 8B8C7
© ':
1
8B$<L7E8N:
/W " 5* L8B8C7N " 8BC<FGH?IJK
*
/X " :5 L8B8C7N " $B*<FGH?IJK
b· õk· LMN ) /·X b· õ·L‚N " z·L‚N
Modal differential equation:
where
%&
' " 7E8<@AB<
t
z· " Ã ã·X LÞN!LÞ; MN HÞ " hij
O
'
A5 •*–
7888 hij E88M
'
Ægˆ<A " 7;<<<<<zW LMN " 7888 hij E88M
Ægˆ<A " *;<<<<<zX LMN " 8
t
b· " Ã 1
© ã·X LÞNHÞ
O
5Þ
*5Þ
8B$ t
X
bW " Ã 8B$ hij
à #7 R fgh
HÞ "
, HÞ " 8B7E' " 8B7EL7E8N " **BE
'
'
* O
O
t
õk· ) /· X õ· "
õW "
wLMN "
7888
hij E88M " ::B:: hij E88M
**BE
::B::
hijLE88MN " 8B8887C hijLE88MN
8BD: R E88X
wLMN " ãW LÞNõW LMN ) ãX LÞNõX LMN ) ¦B
õX LMN " 8;<<<<<2g<fgjhi(…ˆ<gj&*<'iˆh†<$g(…
hij 5 Þ
8B8887C hijLE88MN " 8B88888$C hijLE88MN
7E8
81
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17.7
õk· LMN ) *{· /· õQ· LMN ) /·X õ·L‚N "
t
b· " Ã 1
© ã·X LÞNHÞ
z· LMN
b·
O
bW " **BE
t
z· " Ã 1
©ãLÞN!LÞ; MN HÞ
O
First mode
zW LMN " 7888 hij E88M
/W " 8BC<FGH?IJK
õkW LMN ) * T 8B7 T 8BCõQW LMN ) L8BCNX õW LMN "
Solution
õLMN "
†‡j p "
õW LMN "
zW LMN
bW
õ±‚ hijL/
©M R pN
}L7 R F X NX ) L*{FNX
{ " 8B7;<<<<F "
E88
" D*E
8BC
*{F
*L8B7NLD*EN
"
" R8B888$*
X
7RF
7 R D*EX
:
p " R8B87C3 ;<<<<&W "
5
7888
õ±‚ "
" 7ED*BE
L8BCNX
7ED*BE
hijLE88M ) 8B888$N " 8B88: hijLE88M ) 8B888$N
$U8D*:
wLMN " ãW LÞNõW LMN ) ãX LÞNõX LMN ) ¦B
wLMN " 8B88:
hij 5 Þ
hijLE88M R 8B88$N
'
82
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17.8
=> T h…f X
@AB(
Z
%& " $8 T 78 <=> T @ABX
' " 788<@AB<
& " 7*8<@AB§
1
© " 8BE
/· " $BE7D0
%&
$8 T 78Z L7*8N
" $BE7D0
" *:BC$<FGH?IJK
:
© ':
1
8BE<L788N
õkW LMN ) /WX õW LMN "
t
zW LMN
bW
zW LMN " Ã ãW LÞN!LÞ; MN HÞ
O
Fn ( x) = cosh an x + cos an x - s n (sinh an x + sin an x)
where
sn =
cosh an L - cos an L
sinh an L - sin an L
LG· 'NX " ~· ;<<<<<<45BL7[BE7N‡j(<6‡7&…<7[B:
G· "
Œ$BE7D8
" 8B87U
788
ãW L'N " fghž Œ$BE7D R fgh Œ$BE7D R 8B[$:Lhijž Œ$BE7D R hij Œ$BE7DN
ØW "
fghž Œ$BE7D ) fgh Œ$BE7D
" 8B[$:
hijž Œ$BE7D ) hij Œ$BE7D
ãW L'N " 8B8*:C
zW LMN " 8B8*:C<!LMN
t
bW " Ã 1
© ã·X LÞNHÞ
O
83
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t
t
&W "
0O ãW LÞNHÞ
t
0O ãWX LÞN HÞ
t
" 8B[C$8
à ãW LÞNHÞ " à ƒfghž GW Þ R fgh GW Þ R Ø7 Lhijž GW Þ R hij GW ÞN„HÞ
O
O
ØW "
fghž G7 ' ) fgh G7 '
" 8BE8[
hijž G7 ' ) hij G7 '
LGW 'NX " ~W " $BE7D8<<<<<<
t
à ãW LÞNHÞ "
t
O
GW ' " Œ$BE7D8 " 7BC[E
7
Ïçhijž GW Þ R hij GW ÞçWOO
R 8BE8[çfghž GW Þ ) fgh GW ÞçWOO
O
O Ð
GW
7BC[E
GW "
" 8B87C[E
788
à ãW LÞNHÞ " E$B$ÏL$B7C$[ R 8BUE:7N R 8BE8[L$B$$[7 R 8B*UUE R 7 R 7NÐ " U8BC8
O
t
0O ãW LÞNHÞ U8BC8
X LÞNHÞ
à ãW
"
"
" 77EBUD
8B[C$
&W
O
t
zW LMN " ãW L'N!LMN " 8B8*:C!LMN
õkW LMN ) *UBC$X õW LMN "
!O "
8B8*:C!LMN
" !O \LMN;<<<<<M Ý 8B8E<
77EBUD
8B8*:ULE888N
" 7B8[$\LMN
77EBUD
M¸ 8B8E
"
" 8B*:;<<<<<L°'zN•¿À " 8B[<<<<Lƈg$<Æi8B<:BEN
4W 8B*7
4W "
*5
*5
"
" 8B*7
/W *UBC$
L°'zN•¿À "
õ±‚;W "
õ•¿À;W
õ±‚;W
!O
7B8[$
7
"
"
X
+
*UBC$
C*U
7
" 8B888C:
C*U
õ•¿À;W " ãW LÞ " 'Nõ•¿À;W " 8B8*:C T 8B888C: " *B7 T 78•‰ <@AB
õ•¿À;W " L°'zN•¿À õ±‚;W " 8B[ T
84
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17.9
=> T h…f X
@AB(
%& " 78Ò <=> T @ABX
' " U8<@AB<
1
© " 8BE
wLÞ; MN "
*!O
A5Þ
A5ÞW
A5ÞX
7
L7 R fgh /· MN •hij
l • X hij
)hij
–Ÿ
1
'
'
'
©'
/·
wLÞ; MN "
*!O
A5Þ
A5ÞW
A5ÞX
7
L7 R fgh /· MN •hij
l • X hij
)hij
–Ÿ
1
'
'
'
©'
/·
/· " 5* 0
wLÞ; MN "
At mid section:
17.10
%&
78Ò
" 5* 0
" EB:E<FGH?IJK
:
© ':
1
8BE<LU8N
RD888 7
5Þ
5D8
5[E
L7 R fgh EB:E MN hij #hij
)hij
,
8BELU8N EB:$X
U8
U8
U8
" R:B:UL7 R fgh EB:E MN hij 8B8$EÞL7B$DDN " RDB7$ L7 R fgh EB:E MN hijL8B8$EÞN
hijL8B8$E T :EN " 7B8
wL:E; MN " RDB7$L7 R fgh EB:E MN
õkW LMN ) /WX õW LMN "
t
First mode:
zW LMN
bW
zW LMN " Ã ãW LÞN!LÞ; MN HÞ
O
/W " EB:E<FGH? h…f<<< <ƈg$<9ˆg7&…$<7[BU
t
4W "
*5
" 7B7E<IJK
EB:E
bW " Ã 1
© ãWX LÞNHÞ;<<<<<<<ãW LÞN " hij
O
A5
'
85
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*5
' ÞN< HÞ " 8BE T U8 " **BE
*
*
eO L7 R fgh
bW " 8BE Ã
O
õkW LMN ) EB:EX õW LMN "
R$888
;<<<<<M Ý 8B7<IJK
**BE
õkW LMN ) EB:EX õW LMN " 8;<<<<<M u 8B7<IJK
Max. response from Fig. 4.4:
M
88B7
"
" 8B8CD
4̧ 7B7E
õW;•¿À " 8BEõ±‚ ;<<<<<õ±‚ "
$888
" :BE
**BE T EB:EX
õW;•¿À " 8BEL:BEN " *B*E
at Þ " D8:
at Þ " [E:
wþÿ!;W LÞ; MN " ã7 LÞNõ1GÞ;7 " *B*E hij
õ•¿À;W LÞ " D8:N " 7BUE<@AB
5Þ
U8
õ•¿À;W LÞ " [E:N " 7B7*E<@A
86
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17.11
=> T h…f X
@AB(
%& " $8 T 78Z <=> T @ABX
' " 7C8<@AB<
!LÞ; MN " *888 hij :88M
1
© "7
M¸ "
Modal differential equation for first modes:
õkW ) /W X õW "
/W " **B$[$$0
5
IJK
:88
zLMN 0 !LÞ; MNãW LÞNHÞ
"
;<<<<<<<M Ý M¸
t
bW
© ã X LÞNHÞ
0 1
O
%&
W
$8 T 78Z <
" **B$[$$0
" $B[C<<FGH?IJK
© ':
1
7L7C8N:
õkW ) $B[CX õW "
!O &W
hij /
©M ;<<<<<<<M Ý M¸ ;<<<<&W " 8BC$8C<<<L6‡7&…<7[B$N
©
1
õkW ) $B[CX õW " 8;<<<<<<<M u M¸
õkW ) 7:B*U*õW "
#
*888L8BC$8CN
hij :88M " 7DD* hij :88M<<<<<<<
7B8
4W "
*5
" 7BDD<IJK
$B[C
M
5
"
" 8B88:[
4̧ :88 T 7BDD
õ
" 8B8*E<<<<<Lƈg$<Æi8B<EB$N
,
õ±‚ •¿À;W
õ±‚ "
zO 7DD*
"
" 77DB$
+
7:B*U
õ•¿À;W " 8B8*E<L77DB$N " *BU7
w•¿À;W " ãW LÞNõ•¿À;W
ãW LÞN " fghž GW Þ R fgh GW Þ R Ø7 Lhijž GW Þ R hij GW ÞN
ØW "
fghž G7 ' ) fgh G7 '
hijž G7 ' ) hij G7 '
LGW 'NX " ~W " **B$[$$<<<<<<
87
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GW ' " Œ**B$[$$ " :B[*<<L~W <<<'ˆg$<6‡7&…<7[B$N
ØW " 8BUC$;<<<<<
G7 '
:B*[
" U8G7 "
" *B$D
*
*
ãW LÞ " U83N " 7BECD
17.12
w•¿À;W " 7BECDL*BU7N " :BD7<@AB
Solve for second mode:
/X " D7BD[*C0
%&
1
© ':
" 78B:*<<FGH?IJK
LGX 'NX " ~X
GX ' " ŒD7BD[*C " [BCE<<L~X <<<'ˆg$<6‡7&…<*8B$N
w•¿À;X " ãX LÞNõ•¿À;X
4X "
*5
" 8B$8<IJK
78B:*
M
5
"
" 8B8*D
4̧ :88 T 8B$
õ•¿À;X " 8B7
ãX LÞN " fghž GX Þ R fgh GX Þ R Ø* Lhijž GX Þ R hij GX ÞN
ØX "
fghž G* ' ) fgh G* '
" 7B87:
hijž G* ' ) hij G* '
ãX LÞ " U83N " R8B$E
w•¿À;X " 8B$EL8B7N " 8B8$E<@AB
X
X
w•¿À " rw•¿À;W
) w•¿À;X
" }:BD7X ) 8B8$EX < " :B7D<@AB
88
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19.1
Fourier series is given by Eq. 19.2
;
zLMN " GO ) lLG· fgh Aü
© M ) >· hij Aü
© MN
·+W
4 7 ¡
" Ã zLMNHM
* 4 O
zLMN " $8<<<<<<<<<<<<<8 Ý M Ý
zLMN " R$8<<<<<<<<<<<<<
GO " 8 (area under zLMN between 0 and T=1 sec).
OB‰
G· "
4
*
4
ÝMÝ4
*
* ¡
© M HM
à zLMN fgh Aü
4 O
WBO
G· " * Ã $8 fgh Aü
© M HM ) * Ã R$8 fgh Aü
© M HM
O
OB‰
©
©
D8
Aü
D8
Aü
G· "
© R hij Ÿ
•hij Ÿ R
•hij Aü
Aü
Aü
©
*
©
*
©
D8
Aü
*5
G· "
#* hij
R hij Aü
© ,;<<<<<<<<<<<<<<<<<<ü
©"
" *5
Aü
4
©
*
D8
L* hij 5A R hij *5AN
G· "
*5A
G· " 8<<<<<<<<\PF<G==<AB
OB‰
>· "
* ¡
©M HM
à zLMN hij Aü
4 O
WBO
>· " * Ã $8 hij Aü
© M HM ) * Ã R$8 hij Aü
© M HM
O
OB‰
©
©
D8
Aü
D8
Aü
>· "
R fgh Aü
©Ÿ
•7 R *fgh Ÿ R
•fgh
Aü
Aü
©
*
©
*
©
D8
Aü
*5
>· "
#7 R *fgh
) fgh Aü
© ,;<<<<<<<<<<<<<<<<<<ü
©"
" *5
Aü
4
©
*
D8
L7 R * fgh A5 ) fgh *A5N
>· "
5A
D8
*:8
L7 R *LR7N ) 7N;
<\PF<<<A " 7; $; E; [B B >· "
>· "
5A
5A
D8
L7 R *L7N ) 7N;
\PF<<<A " *; :; D; CB B >· "
<>· " 8
5A
*:8 hij ü
© M hij $ü
© M hij Eü
©M
#
)
)
ÑÑB,
ü
©
7
$
E
7*8 hij *5M hij D5M hij 785M
PF<<<<<<<zLMN "
#
)
)
ÑÑB,
7
5
$
E
Á zLMN "
89
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19.2
From eq. 19.10, the steady-state response of a damped single degree-of-freedom system is:
;
G· *F· { ) >· L7 R F· X N
GO 7
G· L7 R F· X N R >· *F· {
-LMN " ) < Í
hij Aü
©M )
hij Aü
© MÎ
X
X
X
L7
L7 R F· X NX ) L*F· {NX
+ +
R F· N ) L*F· {N
·+W
where<GO ; G· ; >· are the Fourier series coefficients of the function zLMN
In our case,
GO " 8;<<<<<<<<G· " 8
*:8
7*8
>· "
;<<<<<<<<'gˆ<j " 7; $; EB<< #<>· "
<,
A/
A5
©
>· " 8;<<<<<<<<'gˆ<j " *; :; DB<<
F· "
Then,
;
©
+
A/
<<<<<<</ " 0
1
/
>· L7 R F· X N
7
>· *F· {
-LMN " < Í
hij Aü
©M R
hij Aü
© MÎ
L7 R F· X NX ) L*F· {NX
L7 R F· X NX ) L*F· {NX
+
·+W
-LMN "
or in terms of
7*8 T $CD
FGH
/"0
" 7CBU[:*
;<<<<<<<<{ " 8B78
7*CBDD
IJK
©
A/
/
© " *5;<<<<<<<<<<<<<F· "
" 8B$$77A
/
7
L:*BDD:* hij *5M R $B7[$7 fgh *5M ) :B*CU* hij D5M
7*8;888
R D$BC8$8 fgh D5M R :B*$EE hij 785M R 8BC8E[ fgh 785M Ñ N
©
/
" 8B$$77; GAH</
© " *5B
/
7
LL7 R FW X Nhij ü
© M R 8B*FW fgh ü
© MN )
@
C
X
X
X
L7
N
N
R
F
)
8B8:LF
W
W
>
>
>
>
7
X Nhij
7*8
LL7
R
UF
$ü
©
M
R
8B*
T
$F
fgh
$ü
©
MN
W
W
-LMN "
X X
º7 R UFW » ) 8B8:LUFW X N
7*8;8885 ?
B
7
>
>
X
LL7 R *EFW Nhij Eü
>)
© M R 8B* T EFW fgh Eü
© MN ) ¦>
X X
XN
º7
R
*EF
»
)
8B8:L*EF
=
A
W
W
FW "
90
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19.12
a) The Fourier coefficients for the load are found by applying eq. 19.3 with T = 2 sec.
GO "
7 W
!O
à !O hij 5M HM "
* O
5
8<<<<<<<<<<<<<<<<<<A " PHH
* W
à !O hij 5M fgh 5M HM " D!O *
<<<<<A " JoJA
* O
5 7 R 5X
!O
* W
>· " Ã !O hij 5M hij 5M HM " D * <<<<<<<<<<<<<<<<<<A " 7
* O
8<<<<<<<<<<<<<<<<<<<<<A u 7
G· "
The substitution into eq. 19.1 leads to following expression for the Fourier series expansion of the
loading:
or
7 7
*
*
*
!LMN " !O # ) hij ü
©M R
fgh *5ü
©M R
fgh :5ü
©M R
fgh D5ü
©M Ñ Ñ B ,
$5
7E5
$E5
5 *
!LMN " !O L8B$7C$ ) 8BE hij ü
© M R 8B*7** fgh *5ü
© M R 8B8:*:: fgh :5ü
©M
R 8B87C7U fgh D5ü
© M Ñ Ñ B N<<+@]I
where
/
©"
*5
" 5<<LFGH?IJKN
4
b) The steady state response is found applying eq. 19.6.
or
-LMN "
!O 7 C
C
7
©M R
fgh *5ü
©M R
fgh :5ü
© M ) ¦ Ñ B , <<<@AB
# ) hij ü
+ 5 [
7E5
D85
-LMN " !O L8B87C7: ) 8B8DE7$ hij ü
©M ) 8B88UD[E fgh *5ü
© M ) 8B8888$8* fgh :5ü
©M ) ¦ Ñ B N<<<<<@AB
20.1
ƒÚ„ " ¼
:88 R*88
* 8
½<<<<<<<ƒb„ " ¼
½
R*88 *88
8 7
8BE
ƒä„ " ¼ 8BE
½
8B[8[7 R8B[8[7
91
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/W " [BDE<FGH?IJf
/X " 7CB:C<FGH?IJf
~· " ÏãС· ƒb„ l G¥ Lƒb„•W ƒÚ„N¥ Ïãз " l G¥ /·X¥ b· " *{· /· b·
¥
¼
7 /
{
• WŸ " Ë W
{X
* /X
¥
/W( GW
̼ ½
/X( GX
7
8B*
[BDE( Ÿ ¼GW ½
½ " • [BDE
8B7
* 7CB:C 7CB:C( GX
$BC*EGW ) **$BC:UGX " 8B*
UB*:GW ) $7EEBEEDGX " 8B7
GW " 8B8D8CD*[
GX " R8B8887:DE
7 7 8
ƒb„•W " ¼
½
* 8 *
7 7 8 :88 R*88
*88 R788
ƒb„•W ƒÚ„ " ¼
½¼
½"¼
½
R*88 *88
* 8 * R*88 *88
*88 R788
*88 R788 X
½ R 8B8887:DE ¼
½
R*88 *88
R*88 *88
$B$C*E: R8B**D*[
"¼
½
R8B:E*E: $B$C*E:
l G¥ Lƒb„•W ƒÚ„N¥ " 8B8D8CD*[ ¼
¥
ƒ~„ " ¼
* 8 $B$C*E: R8B**D*[
DB[DE R8B:E$
½¼
½"¼
½
8 7 R8B:E*E: $B$C*E:
R8B:E$ $B$C$
92
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20.2
ƒÚ„ " ¼
:88 R*88
* 8
½<<<<<<<ƒb„ " ¼
½
R*88 *88
8 7
8BE
ƒä„ " ¼ 8BE
½
8B[8[7 R8B[8[7
*{W /W bW
8
8
*{X /W bW
ƒE„ " ƒä„¡ ƒ~„ƒä„ " ú
8 <<<<<<<<<<<<<<<8
R <<<<<<<<<<<<<<<R
ê
ƒ~„ " ƒb„ Fl
·+W
8 <<<<<<<<<<<<<<<R
8 <<<<<<<<<<<<<<<<Rû
*{( /W bW R
R
R
*{· /·
Ïãз ÏãС· G ƒb„
b·
ƒ~„ " °G1]@A2<bGMF@Þ
b· " 7B8<@\<APF1G=@õJH
For A " 7
For A " *
ê
*{· /· *L8B*NL[BDEN
"
" $B8D
b·
7
*{W /W
8BE ƒ
8B[DE
7B8C7CD$
ÏãÐW ÏãÐW¡ " ¼
½ 8BE 8B[8[„L$B8DN " ¼
½
8B[8[
78B8C7CD$
7BE$8
bW
*{· /· *L8B7NL7CB:CN
"
" $BDUD
b·
7
*{X /X
8BU*:
R7B$8D[*8C
8BE
ÏãÐX ÏãСX " ¼
½ ƒ8BE R8B[8[„L$BDUDN " ¼
½
R7B$8D[*8C
7BC:C
R8B[8[
bX
l"¼
·+W
{W " *8c " 8B*
{X " 78c " 8B78
8BU*:
R7B$8D[*8C
8B[DE
7B8C7CD$
7BDCU
R8B**:CE[C
½)¼
½ "¼
½
R7B$8D[*8C
7BC:C
78B8C7CD$
7BE$8
R8B**:CE[C
$B$[C
ê
*{· /·
* 8
7BDCU
R8B**:CE[C * 8
Ïãз ÏãС· G ƒb„ " ¼
½¼
½¼
½
8 7 R8B**:CE[C
8 7
$B$[C
b·
·+W
$B$[C
R8B::U[* * 8
DB[ED R8B:E8
"¼
½¼
½"¼
½
R8B**:CE[C
$B$[C
8 7
R8B:E8 $B$[C
ƒ~„ " ƒb„ Fl
93
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[888 R$888
8
ƒÚ„ " åR$888 E888 R*888æ
8
R*888 *888
7E 8 8
ƒb„ " å 8 78 8æ
8
8 E
8B777: R8B7UDC R8B7*:E
ƒä„ " å8B*77[ R8B8*[[ 8B*$$$ æ
8B*[8$ 8B*CDC R8B*77:
20.3
/W " UB$7<FGH?IJf
/X " *8BU:<FGH?IJf
/( " *UB88<FGH?IJf
{W " 78c " 8B78
ê
ƒ~„ " ƒb„ Fl
·+W
For A " 7
*{· /·
Ïãз ÏãС· G ƒb„
b·
*{· /· *L8B7NLUB$7N
"
" 7BCD*
b·
7
8B8*$ 8B8:$ 8B8ED
8B77:
*{W /W
ÏãÐW ÏãÐW¡ " Ë8B*77[Ì ƒ8B77: 8B*77[ 8B*[8$„L7BCD*N " å8B8:: 8B8C: 8B78[æ
bW
8B*[8$
8B8ED 8B78[ 8B7$D
For A " *
*{· /· *L8B7NL*UB88N
"
" :B7CC
7
b·
8B7D*
8B8** R8B*$D
R8B7UDC
*{X /X
ÏãÐX ÏãСX " Ë 8B8*[[ Ì ƒR8B7UDC 8B8*[[ 8B*CDC„L:B7CCN " å 8B8*$
8B88$ R8B8$$æ
bX
8B*CDC
R8B*$D R8B8$$ 8B$:::
For A " $
*{· /· *L8B7NL*UB88N
"
" EBC
b·
7
R8B8U8 R8B7DC 8B7E$
R8B7*:E
*{( /(
ÏãÐ( ÏãС( " Ë 8B*$$$ Ì ƒR8B7*:E 8B*$$$ R8B*77:„LEBCN " åR8B7DC 8B$7D R8B*CDæ
b(
R8B*77:
8B7E$ R8B*CD 8B*EU
8B*[E R8B78* R8B8*C
l " åR8B78* 8B:8* R8B*7$æ
R8B8*C R8B*7$ 8B[:8
·+W
ê
7E 8 8 8B*[E R8B78*
*{· /·
Ïãз ÏãС· G ƒb„ " å 8 78 8æ åR8B78* 8B:8*
b·
8
8 E R8B8*C R8B*7$
·+W
:B7*C R7B87[ R8B7$C 7E 8 8
D7BU*
" åR7BE*D :B8*: R7B8D:æ å 8 78 8æ " åR7EB*D
8
8 E
R*B8C8
R8B:7E R*B7*C $BDUU
ê
ƒ~„ " ƒb„ Fl
R8B8*C 7E 8 8
R8B*7$æ å 8 78 8æ
8B[:8
8
8 E
R7EB*D R*B8C8
:8B*: R78BD:æ
R78BD: 7CBE8
94
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21.1
*E88\ LMN
ªLMN
$
ªQ T K
*
1ªk
&ªk?*:
$+ T ª
The figure shows the forces acting in the system for a displaced position of the generalized coordinates,
ªLMN
Give virtual displacement vª to generalized coord. ªLMN and apply principle of virtual work:
vª
& T ªk T vª $
$
R1 T ªk T vª R
R ª Q T K vª R $+ T ª T $vª R *E88\LMN
"8
*: T 7*
*
*
:
Since vª H 8
Generalized Parameters:
bI "
&
U
*E88
) 1Ÿ ªk R KªQ R U+ª "
\LMN
•
:
:
*CC
&
7 7 C88
C88
=> T IJK*
)1 "
LD8NX )
" :B*$<<
*CC
*CC 7* $CD
$CD
@AB
U
U
=> T IJK
~ I " K " 788 " **E<
:
:
@AB
Ú I " U+ " ULE888N " :E888<=>?@AB
z I LMN "
*E88
\LMN " D*E\LMN<=>B
:
95
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21.2
The figure shows the system in displaced position and corresponding forces:
&| T ªk?G
+ª
ªLMN
1 T ªk
ªQ K
&^ ªk ?'
bLMN
Give virtual displacement vª and apply principle of virtual work:
Since vª H 8
R
1 T ªk T vª &^ T ªk T vª
&| T ªk T vª
vª
R
R ªQK T vª R + T ª T vª R
) bLMN
"8
*
*
'T'
GTG
'
#
Generalized parameters:
bI "
1 &^ &|
bLMN
) X ) X , ªk ) K T ª ) + T ª "
* '
G
'
1 &^ &| 1 7 *1'X 1GX [
) )
" )
)
" 1
* 'X GX * 7* 'X
*GX D
~I " K
ÚI " +
bLMN
z I LMN "
'
96
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21.3
The figure shows the system in displaced position for the generalized coordinate pLMN:
KpQ T '
1
© T 'X T pk
*
& T pk
!O T ' T \LMN
'
*
$
pLMN
+T'Tp
1
© T 'X
pk
*
& T pk
pLMN
Give virtual displacement<vp and apply Principle of virtual work:
1
© T 'X T pk T vp'
!O T ' T \LMN '
R* V
Y R *º& T pk T vp» R KpQ T ' T ' T vp R + T p T 'vp R
T vp " 8
*
$
*
*
Since vp H 8
Generalized parameters:
1
© T 'X &
!O T ' T \LMN
V
) Y pk ) K T ' T pQ R + T ' T p "
'
D
*
bI "
© T 'X
© T ' T 'X
1
7 1
[
)
"
1
© 'X
7*
7*
*
'
~ I " K'
Ú I " +'
!O T ' T \LMN
z I LMN "
D
97
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21.4
Generalized mass:
t
1
bI " 1 ) Ã • – ã X LÞN HÞ
'
O
bI " 1 )
bI " 1 )
bI " 1 )
1 t
5Þ X
à •7 R fgh – HÞ
' O
*'
1 t
5Þ
5Þ
à •7 R * fgh
) fghX – HÞ
' O
*'
*'
1
:'
5Þ Þ
'
5Þ t
) )
hij Ÿ
•Þ R hij
'
5
*' * *5
' O
1 $
:'
# 'R ,
' *
5
1
LE5 R CN
bI "
*5
bI " 1 )
t
Generalized stiffness:
Ú I " Ã %&ã:X LÞN HÞ
O
5 X
5Þ
5Þ
ãLÞN " 7 R fgh ;<<<<<<<<<<ã:LÞN " • – fgh
*'
*'
*'
t
Ú I " %&
Generalized force:
Ú I " Ã %& •
O
5 §
5Þ
HÞ
– fghX
*'
*'
5 *' 5Þ
*' 5Þ
5Þ t %&5 §
)
hij
•
•
–
•
–
Ÿ "
7D'§ 5 :'
5 :'
' O $*'§
§
'
z I LMN " Ã ]LÞ; MNãLÞN HÞ
z
z
I LMN
I LMN
z
8
'
" zO \LMNã # ,
*
5
" zO •7 R fgh – \LMN
I LMN
:
" 8B*U*U T zO \LMN
98
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21.5
The generalized geometric stiffness:
t
Hã X
I
Ú. " J Ã # , HÞ
HÞ
O
ãLÞN " 7 R fgh
5Þ Hã
5
5Þ
;<<<<<
"
fgh
*'
HÞ *'
*'
t
5 X
5Þ
Ú.I " RJ Ã • – hijX
HÞ;<<<LAJ2GM@oJ<I@2A<>JK-GIJ<J<@I<MJAI@PAN
*'
*'
O
7
5Þ t
J5 X
5 X *' 5Þ
5 5
Ú.I " RJ • –
•• – R hij Ÿ " RJ • – " R
:
' O
C'
*' 5 :'
*' :
ÚI "
ÚaI " Ú I R Ú.I
%&5 §
<<<<<\FP1<!FP>=J1<*7B:<
$*'(
%&5 § J5 X
ÚaI "
)
$*'(
C'
99
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21.6
The generalized mass:
t
bI " Ã 1LÞNã X LÞN HÞ
O
K 5HX X
K
HÞ "
Þ HÞ
2 :'X
2
5Þ
ãLÞN " 7 R fgh
*'
K 5HX t X
5Þ X
I
b "
Ã
Þ
R
fgh
•7
– HÞ
2 :'X O
*'
1LÞNHÞ " ¯À
K 5HX '(
*' ( 5 X
'( '(
b "
R
*
#
,
#•
)
R YÌ
R
*,
Ë
–
V
2 :'X $
5
*
D 5X O
I
bI "
The generalized stiffness:
t
K 5H 7 : $* 7
KH
# R ) ( R X , " 8B7*$[
2 : * 5 5
5
2
t
Ú I " Ã %Š &LÞNã:X LÞN HÞ;<<<<<<&LÞN "
O
5 HX Þ §
: 7D '§
5Þ
5HX 5 X t §
5 X
5Þ
Ú I " %Š
• – Ã Þ fgh HÞ;<<<<<ã:LÞN " • – fgh
X
*'
D:' *'
*'
*'
O
ÚI "
The generalized force is
z
I LMN
H
" ]8 LMN Ë
'
'*
*
z
R
'
%Š 5HX
7*C'X
z I LMN " Ã ]LÞ; MNãLÞN HÞ
I LMN
"Ã
:'* 5
8
'
ÞH
'
8
]8 LMN •7 R fgh
• R 7–Ì "
*
5
*
]8 LMN' T H
*5* '
5Þ
*'
– HÞ
ƒU R :5„ " R8B7C8[]8 LMN' T H
100
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21.7
The deflection occurred corresponding to the left half a simply supported beam with a concentrated
load at its center is
wO
w " ( L$'X Þ R :Þ ( N
'
Where wO is the deflection at the center.
The maximum potential energy is calculated as
7
É•¿À " zwO
*
*:%& X
z'(
É•¿À " ( wO ;<<<<<I@AKJ<wO "
'
:C%&
and the maximum Kinetic Energy
'?*
4•¿À " * Ã
4•¿À "
4•¿À "
8
1> º/w8 »
'
1> º/w8 »
'
'
D
4•¿À "
*
'
D
*
7 1>
*
•
7
*
HÞ– L/wN* ) 1º/w8 »
'
*
'?*
7
à L$'* Þ R :Þ$ NHÞ ) 1º/w8 »
'
8
$
å$': # , )
*
1> º/w8 »
[TCTE
*
7D '
:
# , R
[ *
*:'
E
*
*
'
E
7
*
# , æ ) 1º/w8 »
7
*
*
ƒ78E ) E R :*„ ) 1º/w8 »
*
É•¿À " 4•¿À
*:%& X
7[
* 1
wO " º/w8 » # ) 1> ,
(
'
* [8
:C%&
/"0
7[
'( •1 ) 1> –
$E
*
*
101
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21.8
The beam loaded as shown in Fig. (a), between supports is equivalent to beam shown in Fig. (b)
!
bO "
!
*
LGN
Moment a section Þ in Fig (b):
bÀ "
w"
Þ " 8;
Max. potential:
Max. Kinetic Energy:
4•¿À "
w"
$!
*
!Þ
HX w
!Þ
Hw !Þ X
¶<<<<<<< X "
¶<<<<<<<
"
) ~W
*
HÞ
*%&
HÞ *%&
!Þ (
) ~W Þ ) ~X ;
7*%&
Deflection of overhang:
at <
L>N
!'
*
GM<Þ " 8;<<<w " 8;<<<~X " 8<
!'(
) ~W '
7*%&
X
!'
Hw
!'X
~W " R
;<<<<<<< â
"
7*%&
HÞ À+t D%&
!Þ (
!'X
!
ƒÞ ( R 'X Þ„
w"
R
Þ"
7*%& 7*%&
7*%&
Þ " ';<<<w " 8 "
HX w
! '
" # R Þ,
X
HÞ
%& *
ÞX
Hw ! '
" V Þ R Y ) ~(
*
HÞ %& *
! ' X Þ(
w " V Þ R Y ) ~( Þ ) ~§
%& :
D
Hw !'X
"
" ~( ;<<<Þ " 8;<<<w " 8; <<<~§ " 8<
HÞ D%&
!
L$'Þ X R *Þ ( ) *'X ÞN;<<<<@A<PoJFLGA2
7*%&
!wO
*
!
'X
'(
'
wO "
R
*
) *'X Y
V$'
7*%&
:
C
*
!X '(
!'(
É•¿À "
;<<<wO "
<<
7D%&
C%&
É•¿À "
7 1x /X t
7 1x /X t?X X
1x /X !X 'Ò
!X
Z
§ X
X § NHÞ
LÞ
Ã
)
'
Þ
R
*'
Þ
)
Ã
w
HÞ
"
ƒ8B8[D ) 8B$D7E„
* $' O 7::L%&NX
* $' O
$D T 7::L%&NX
*
*
4•¿À " É•¿À
102
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1x /X !X 'Ò !'(
"
UC[L%&NX
C%&
%&
/ " [BCE:0 §
' 1>
103
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21.9
The deflection at section x is:
Maximum velocity is:
w " ªhij
5Þ
hij /M
'
wQ•¿À " ª/ hij
5Þ
'
t?X
7
7
7 1x X ' /X ª X
7 1x X X 5Þ
L*1 ) 1x N " E/X ª X
4•¿À " 1ª X /X ) Ã
ª / hij HÞ " 1ª X /X )
ª "
*
'
*
*
*
'
'
:
*
O
Maximum Potential Energy:
X
%& t?X HX w
HÞ
É•¿À " Ã V X Y
HÞ •¿À
* O
and Max kinetic Energy:
É•¿À "
5
5Þ
HX w
5X
5Þ
Hw
" ª fgh hij/M;<<< X " Rª X hij hij/M
'
'
HÞ
'
'
HÞ
%& t?X X 5 § X 5Þ
%& 5 § ' 789 ª X 5 §
à ª § hij
" *:$Eª X
HÞ " ª X § "
* O
'
'
' *
: 788(
*
4•¿À " É•¿À
E/X ª X " *:$Eª X
/X " :C[
/ " **B8[<FGH?IJK
**B8[
\"
" $BE7<K]I<
*5
104
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21.10
Use Eq. 21.83:
/"0
2 y M¥ w¥
y M¥ w¥ X
In this equation the deflection w¥ at the floor levels is due the static load W applied horizontally at each
story.
The shear force in the first story is:
ÉW "
And in the second story:
Since ÉW " *3 and ÉX " 3
ÉX "
7*%&
w
'( W
7*%&
LwX R wW N
'(
M'(
7*%&
M'(
M'(
$M'(
wX " *
)
"
7*%& 7*%& 7*%&
wW " *
M'( $M'(
EM X '(
)
Ì"
7*%& 7*%&
7*%&
X 'Z
X 'Z
M
UM
7$M ( 'Z
l M¥ w¥ X " <M Ë:
)
Ì
"
L7*%& NX L7*%& NX
L7*%&NX
l M¥ w¥ " M Ë*
EM X '(
2L7*%&NE :BD*%&
/X " 7*%&
"
"
(
Z
7$M '
M'(
7$M'(
X
L7*%&N
2
/ " *B7E0
2%&
M'(
105
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21.11
As in problem 21.6 load the frame with horizontal force W at each story.
For fixed column the shear force is
É"
Then
For the first story: É " *3;<<<<<<<×" wW
×"
¯N×
'
É'
¯N
*3'
wW "
¯N
3' $3'
wX " wW )
"
¯N
¯N
2 y M¥ w¥
/"0
y M¥ w¥ X
M' $M'
EM X '
)
Ÿ"
¯N ¯N
¯N
X 'X
X 'X
M
UM
7$M ( 'X
l M¥ w¥ X " <M Ë:
)
Ì
"
L¯N NX L¯N NX
L¯N NX
l M¥ w¥ " M •*
EM X '
2 ¯N
EU2¯N
/X "
"
(
X
7$M '
7$M'
L¯NNX
/ " 8BD*0
2¯N
M'
106
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21.12
The deflection shape of a cantilever beam with a concentrated load at its end is:
ª•¿À
L$'Þ X R Þ ( N
*'(
where ª•¿À is the maximum deflection at the free end
wLÞN "
Maximum Potential Energy:
É•¿À "
Hw ª•¿À
LD'Þ R $Þ X N
"
HÞ
*'(
$ª•¿À
HX w ª•¿À
LD' R DÞN "
L' R ÞN
"
(
X
*'
'(
HÞ
%& Uª•¿À X
%& Uª•¿À X t?X X
%& t?X HX w
7
XN
L'
HÞ
"
à V X Y HÞ "
Ã
R
*'Þ
)
Þ
#7 R 7 ) , " EEC7BDEª•¿À X
Z
*
'
HÞ
* 'Z
* O
$
O
X
7
1
1
©wQ X HÞ ) OwQW X ) wQ X X ) wQ ( X P
*
O *
78/* ª•¿À X t X §
7888/* '$ C *C
‰
ZN
LU'
4•¿À "
Ã
Þ
R
D'Þ
)
Þ
) *Ÿ ª•¿À X
HÞ
)
• )
Z
C'Z
C'
*[
*[
O
78/* ª•¿À X ' U D 7
7888/* '$
4•¿À "
R
)
)
$B$$ª•¿À X
•
Ÿ
C'Z
C'Z
E D [
*
" E8UB7:/* ª•¿À X ) :7BD[/ ª•¿À X " E8EBC7/* ª•¿À X
Maximum Potential Energy:
t
4•¿À " Ã
4•¿À " É•¿À
EE8BC7/ ª•¿À X " EEC7BDEª•¿À X
/X " 78B7$
/ " $B7C<FGH?IJK
\ " 8BE8[<K]I<
X
107
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21.3
The deflection shape of a cantilever beam with a concentrated load at its end is:
and the deflection at a section x by:
Maximum Potential Energy:
É•¿À "
7 Þ§ :
D
ãLÞN " V § R ( Þ ( ) X Þ X Y
$ '
'
'
wLÞN " ª•¿À ãLÞN hij /M
7*
7 :Þ ( 7*
Hw
" ª•¿À V § R ( Þ X ) X ÞY hij /M
'
'
$ '
HÞ
HX w
7*
7 7*Þ X *:
" ª•¿À V § R ( Þ ) X Y hij /M
HÞ X
'
'
$ '
ª•¿À X %& t 7::Þ § E[D X 7:: E[DÞ ( *CC X E[DÞ
%& t HX w
HÞ "
à V XY
à V 9 ) Z Þ ) § R
) Z Þ R ‰ Y HÞ
7C
* O HÞ •¿À
'
'
'Ò
'
'
'
O
X
É•¿À "
Max. Kinetic Energy:
7::ª•¿À X %& 7 :
: * :
7::%&ª•¿À X
" EUE$B[:ª•¿À X
• ) )7R ) R Ÿ"
U8
7C
: $ *
E $
$ T 78WW
%&
"
" $[*7B7
(
L$D T 7*N(
'
1
7
1
©wQ X HÞ ) OwQW X ) wQ X X ) wQ ( X P
*
O *
78/* ª•¿À X t Þ 9 7DÞ Z $DÞ § CÞ Ò 7*Þ Z :CÞ ‰
788/*
4•¿À "
Ã
R
)
R
)
R
7*BC8$ª•¿À X
HÞ
)
Y
V
7C'Z
'Z
'§
'Ò
'Z
'‰
7C
'9
O
78/* 'ª•¿À X 7 7D $D C 7* :C
X
4•¿À "
)
R )
R Ÿ ) [7B7$/* ª•¿À
• R
Z
7C'
E C [
D
U [
X
4•¿À " L*:8 ) [7B7$N/* ª•¿À
t
4•¿À " Ã
----------------------------------------------------------------------END--------------------------------------------------------------------------
Prepared by Mario Paz and Young Hoon Kim
108
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http://www.springer.com/978-3-319-94742-6
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