Solution Manual To Structural Loads 2012 IBC And ASCE SEI 7-10 2.pdf

Solution Manual for c c




Structural Dynamics: The
c c



ory and Computation
c c




Sixth Edition c




Mario Paz and Young Hoon Kim
c c c c c

,1.1


Ifcthecweightcwciscdisplacedcbycamount,cy,cthecbeamcandcthecspringscwillcexertcactotalcforceconc
thecmasscof
3𝐸𝐼c
𝑃 c =c( +c2𝑘)c𝑢
𝐿3

Thecbeamcandcspringscactcincparallel.cThecequivalentcstiffnesscis:
𝑃c 3𝐸𝐼c
𝑘c = c = c ( +c2𝑘)
𝑒
𝑢 𝐿3

Naturalcfrequency:

𝑘 𝑔 3𝐸𝐼
𝜔c =c√
c c
=c√ ( +c2𝑘)
𝑚 𝘸 𝐿3

Naturalcperiod:

2𝜋c 𝘸c 𝐿
𝑇c =c =c2𝜋𝐿√ ( )
𝜔 𝑔 3𝐸𝐼c+c2𝑘𝐿 3



1.2


Stiffness:

3𝐸𝐼 3c×c108
𝑘𝑒c =c (c +c2𝑘)c= +c2c×c1000c =c 4,300c𝑙𝑏/𝑖𝑛.
𝐿3c c 1003


Naturalcfrequency:
√ c𝑘 4300c×c386
𝜔c = =c√ =c 23.52c𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚c 3000

Freecvibrationcresponsecofcundampedcoscillator:
𝑢˙Oc
𝑢(𝑡)c =c 𝑢O 𝑐𝑜𝑠𝜔𝑡c+c 𝑠𝑖𝑛𝜔𝑡
𝜔


1

, 𝑢˙c(𝑡)c=c−𝑢O𝜔𝑠𝑖𝑛𝜔𝑡c+c𝑢˙O𝑐𝑜𝑠𝜔𝑡

Displacementcandcvelocitycatc𝑡c =c 1𝑠𝑒𝑐c withcthecinitialcvalues

𝑢Oc=c1c𝑖𝑛.c,c𝑢˙Oc =c20c𝑖𝑛./𝑠𝑒𝑐:
20
𝑢(1)c =c 1c∙c𝑐𝑜𝑠(23.5c∙c1)c+c 𝑠𝑖𝑛(23.5c∙c1)c =c −0.89c𝑖𝑛.
23.5
𝑢˙c(1)c=c−1c∙c23.5𝑠𝑖𝑛(23.5c∙c1)c+c20𝑐𝑜𝑠(23.5c∙c1)c=c22.66c𝑖𝑛./𝑠𝑒𝑐




1.3

Thecstiffnesscofcthecbeamcis


12𝐸𝐼1 3𝐸(2𝐼2) 12c∙c(30c∙c106)c∙c170.9 3c∙c(30c∙c106)c∙c82.5
𝑘c =c Vc c + Yc = + =c 25,577c𝑙𝑏/𝑖𝑛.
𝐿3 𝐿3 1443 1443


Naturalcfrequency:

1c 𝑘 1 25,577c×c386
√
𝑓c =c =c √ =c 2.24c𝑐𝑝𝑠
2𝜋 𝑚c 2𝜋c 50,000




1.4

a) Infinitelycrigidchorizontalcmemberc

Stiffness:

12𝐸𝐼 12c∙c(30c∙c106)c∙c171
𝑘c =c 2c(c 3c c )c = =c 21,100c𝑙𝑏/𝑖𝑛.
𝐿 (12c∙c15)3


Naturalcfrequency:



2

, √ c𝑘 21,100c×c386
𝜔c = =c√ =c 18.05c𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚c 25,000
𝜔
𝑓c =c =c 2.87c𝑐𝑝𝑠
2𝜋

b) FlexiblechorizontalcmembercconsistingcofcW18X30

Computecthecstiffnesscbycmomentcdistributioncmethod.cDisplacecthecframechorizontallycbyco
necinchcandcdeterminecthecstiffnesscofcthecframecascthecsumcofcthecshearcforcescincbothccolu
mns.cTakecadvantagecofcthecsymmetrycbycmodifyingcthecstiffnesscofchorizontalcmembercbycfac
torc3/2.


Distributioncfactors
4𝐸𝐼c 4𝐸c
𝑘 =c =c ∙c171c →c 171c →c 0.1244
𝐵𝐴
𝐿 𝐿
4𝐸𝐼c3c 4𝐸c c 3
𝑘 =c =c ∙ ∙c802c →c 1203c →c 0.8756
𝐵𝐶
𝐿c c 2 𝐿c 2c


Fixedcendcmoments:
0.1244 0.8756
C
950 B
6𝐸𝐼 6c∙c(30c∙c106)c∙c170.9 -118 -832
𝑀𝐵𝐴 =c 𝑀𝐴𝐵 = = (12c∙c15)2 c c -832c
𝐿2
832
=c 950c(𝑘c −c𝑖𝑛.c)

Shearcforce:
832c+c891c
𝑘c =c ∙c2c =c 19.14c𝑘𝑖𝑝/𝑖𝑛.
180 950 A D
-59
Naturalcfrequency:
891

1c 𝑘 1 19,140c×c386
√ =c √
𝑓c =c =c 2.74c𝑐𝑝𝑠
2𝜋 𝑚c 2𝜋c 25,000



Note:
1. Assumingcacflexiblecgirdercdecreasescthecnaturalcfrequencycbyconly


3