Lecture 02- DVA Spectra and Generalized SDOF Systems.pdf

Engineering dynamics lecture

1. CEE 424-Elementary Structural Dynamics
Lecture 02-DVA Spectra and Generalized SDOF Systems
Semester –January 2020
Dr.TahirMehmood

2. Combined D-V-A Spectrum:
Eachofthedeformation,pseudo-velocity,andpseudo-accelerationresponsespectrafora
givengroundmotioncontinuethesameinformation-theyaresimplydifferentwaysof
presentingthesameinformationonstructuralresponse.
7.48=√7�8

3. A single curve can simultaneously show three different quantities.
•The peak deformation
•The peak pseudo-velocity which is related to the peak strain energy
•The peak pseudo-acceleration which is related to the peak value of
equivalent static force (and base shear).

5. Example
A12-ft-longverticalcantilever,a4-in.-nominal-diameterstandardsteelpipe,supportsa
5200-lbweightattachedatthetipasshowninFig.E6.2.Thepropertiesofthepipeare:
outsidediameter,�
�=4.500in.,insidediameter�
�=4.026in.,thickness�=0.237in.,
andsecondmomentofcross-sectionalarea,??????=7.23in4,elasticmodulus�=29,000ksi,
andweight=10.79lb/footlength.Determinethepeakdeformationandbendingstressinthe
cantileverduetotheElCentrogroundmotion.Assumethat??????=2%.
Solution

7. The lateral stiffness of this SDF system is
�=
3�??????
�
3
=
329×10
3
7.23
12×12
3
=0.211Τ�??????????????????�.
The total weight of the pipe is 10.79×12=129.5��, which may be neglected relative to the
lumped weight of 5200��. Thus
�=
�
�
=
5.20
386
=0.01347�????????????− Τ���
2
??????�.

8. From the response spectrum curve for ??????= 2% (Fig. E6.2b), for ??????
�= 1.59 sec, �= 5.0 in. and
�= 0.20g. The peak deformation is
�
�=�=5.0??????�.
The natural vibration frequency and period of the system are
�
�=
�
�
=
0.211
0.01374
=3.958 ������??????
�=1.59���
w=2??????/??????�

9. Note:The unit of force is �????????????,
1 kip = 1000 ��
The unit of mass is therefore the unit of force divided by the unit of acceleration.
The unit of acceleration is ??????�/sec
2
1�=9.81�/sec
2
=32.2��/sec
2
=
386??????�/sec
2
The unit of �is the unit of force divided by the unit of area. The lateral stiffness �in
this case is determined from
�=
�
??????

10. The peak value of the equivalent static force is
�
??????�=
�
�
�=0.20×5.2=1.04�????????????�
The bending moment diagram is shown in Fig. E6.2d with the maximum moment at the base =
12.48 kip-ft. Points A and B shown in Fig. E6.2e are the locations of maximum bending stress:
??????
��??????=
�
�
??????
=
(12.48×12)(Τ4.52)
7.23
=46.5��??????
Asshown,??????=+46.5ksiat�and??????=−46.5ksiat�,where+denotestension.Thealgebraic
signsofthesestressesareirrelevantbecausethedirectionofthepeakforceisnotknown,as
thepseudo-accelerationspectrumis,bydefinition,positive.

11. Response Spectrum Characteristics
Combined D-V-A response spectrum (??????=0, 0.02, 0.05, 0.1) and peak values of ground
acceleration, ground velocity, and ground displacement for El Centro ground motion

12. •Forsystemswithveryshortperiod,thepseudo-acceleration�foralldampingvalues
approachሷ�
??????0and�isverysmall.
•Forsystemswithverylongperiod,�forallthedampingvaluesapproach�
??????0and�is
verysmall;thustheforcesinthestructures,whicharerelatedtothe��,wouldbevery
small.
•Thereductionofresponseduetoadditionaldampingisdifferentforthedifferentspectral
regions-greatestinthevelocity-sensitiveregion.(Theeffectivenessofdampinginreducing
thestructuralresponsealsodependsonthegroundmotioncharacteristics).
•ForMexicocity85earthquakewheregroundmotionisnearlyharmonicovermanycycles,
theeffectofdampingwouldbelargeforasystemnear“resonance”.
•ForParkFiled66earthquakewheregroundmotionisveryshortandshocklike,theeffect
ofdampingwouldbesmall,asinthecaseofhalfcyclesinepulseexcitation.

13. Averyshortperiodsystemis
extremelystiffandrigid.Its
deformationresponsetotheground
motionisverysmall.Soitsmass
moverigidlywiththegroundandits
peakstructuralaccelerationshould
beapproximatelyሷ�
??????0.
Todrivethestructuralmasstomove
withaccelerationofሷ�
??????0,itis
necessarytohave�
??????0≈�ሷ�
??????0,
therefore,�≈ሷ�
??????0

14. Averylongperiodsystemisextremelyflexible.Themasswouldbeexpectedtoremain
essentiallystationarywhilethegroundbelowmoves.
Thus ��≅−�
??????�thatis �≅�
??????0

15. Generalized SDOF systems:
Considering 3 examples of complex systems

16. If the deformation of complex systems can be (approximately) expressed as:
��,�=��??????� (1)
where ��: a shape function (dimensionless) or a mode shape
??????�: a single generalized displacement
or, for the case (c),
�
��=�
�??????� (2)
where �
�: a shape factor (dimensionless) at the ??????
�ℎ
story

17. Then it can be shown that the governing equation of motion of each of these complex
systems is the form of
෥�ሷ??????�+ǁ�ሷ??????�+෨�??????�=−෨�ሷ�
??????� (3)
where෥�,ǁ�,෨�,෨�are defined as the generalized mass, generalized damping,
generalized stiffness and generalized force factor of the system, respectively.

18. ෨�=�
���
�+�
���
�
For the system “a”:
෥�=�
��
2
�
�+�
��
2
�
�
ǁ�=�
��
2
�
�
෨�=�
��
2
�
�+�
��
2
�
�
(4)
For the system “b”:
(5)
෨�=න
0
??????
�??????��
′′
�
2
��
෥�=න
0
??????
���
2
���
෨�=න
0
??????
������

19. For the system “c”:
(6)
෥�=෍
�=1
??????
�
��
2
�
(N=4 in this case)
෨�=෍
�=1
??????
�
�(�
�−�
�−1)
2
where ; ℎ: story height
෨�=෍
�=1
??????
�
��
�
Therefore,
Given the vibration shape and the distribution of mass and flexibility of a complex system, it is
possible to evaluate all of these generalized properties of the system.
�
�=෍
���??????��
12�??????
ℎ
3

20. Dividing the Eq. (3) by ෥�gives:
ሷ??????+2??????�
??????ሶ??????+�
2
�??????=−෨Γሷ�
??????� (7)
where
෨Γ=ൗ
෨??????
෥�
: (dimensionless factor)
�
�
2
=ൗ
෨�
෥�
: the natural frequency of the generalized SDOF system
??????=ൗ
ǁ�
2෨�෥�
: the (model) damping ratio
(7
′
)
Theequation(7)saysthatthegeneralizeddisplacement??????�ofthegeneralizedSDOFsystem
duetogroundmotion�
??????�isidenticaltothedisplacementresponse��ofasimpleSDOF
system(havingthesystem�
�and??????)togroundmotion෨Γ�
??????�.

21. Suppose that the response spectrum of the ground motion �
??????�is available. It is then possible
to estimate peak earthquake response of the generalized SDOF system:

22. Peak value of ??????�=??????
0=෨Γ�=
෩Γ
??????
2
??????
� (8)
where �and �are the deformation and pseudo-acceleration ordinates, respectively, of
the spectrum at the modal period ??????
�=Τ
2??????
????????????
for the modal damping ratio ??????
Peak displacements �
0�=෨Γ���
�
�0=෨Γ��
�
or
(9)
(10)
Equivalent static forces (external static forces that would cause displacement �
0�for the
system “b” can be derived from elementary beam theory as

23. �
??????0�=[�??????��
0
′′
�]
″
or alternatively
�
??????0�=���
2
��
0�=෨Γ�����
(11)
(12)
Thus, the shear and bending moment of height �above the
base are:

24. ??????
0�=න
??????
??????
�
??????0??????�??????=෨Γ�න
??????
??????
�??????�??????�?????? (13)
�
0�=න
??????
??????
(??????−�)�
??????0??????�??????=෨Γ�න
??????
??????
(??????−�)�??????�??????�?????? (14)
The shear and bending moment at the base of the tower are
??????
�0=??????
00=෨�෨Γ�
�
�0=�
00=෨�
??????෨Γ�
෨�
??????
=න
0
??????
�������
(15)
(16)
(17)where

25. For the case of “shear building” (system “c”)
�
�0=෨Γ��
��
�
for j=1,2,3,……,N
In this case N=4
(18)

26. The overturning moment �
�0at the ??????
�ℎ
story is :
�
�0=෍
�=�
??????
(ℎ
�−ℎ
�)�
�0=෨ΓA෍
�=�
??????
(ℎ
�−ℎ
�)�
��
�
The shear force ??????
�0in the ??????
�ℎ
story is:
??????
�0=෍
�=�
??????
�
�0=෨Γ�෍
�=�
??????
�
��
� (19)
(20)
where ℎ
�is the height of the ??????
�ℎ
floor above the base.

27. The shear and overturning moment at the base are
??????
�0=෍
�=�
??????
�
�0=෩�෩Γ� (21)
�
�0=෍
�=�
??????
ℎ
��
�0=෨�
??????෨Γ�
where ෨�
??????
=෍
�=�
??????
ℎ
��
��
�
(Noteherethatthegeneralizedfactors෨�,෨�
??????
and෨Γdependonlyonthevibrationshapeand
themassdistributionofthecomplexsystems.)
(22)
(23)

28. Thank you