Participation factorsParticipation factors
7. Response Spectrum Analysis Response Spectrum Analysis
7. Response Spectrum Analysis
Description Description
Response spectra are plots of maximum response of single degree of freedom Response spectra are plots of maximum response of single degree of freedom (SDOF) systems subjected to a specific excitation. For various values of frequency (SDOF) systems subjected to a specific excitation. For various values of frequency of the SDOF system and various damping ratios, the peak response is
of the SDOF system and various damping ratios, the peak response is calculated.calculated.
Structures normally have multiple degrees of freedom (MDOF). The dynamic Structures normally have multiple degrees of freedom (MDOF). The dynamic analysis of a
analysis of a MDOF system having "n" DOF involves reducing it to MDOF system having "n" DOF involves reducing it to "n" independent"n" independent SDOF systems. The modal superposition method is used and the maximum modal SDOF systems. The modal superposition method is used and the maximum modal responses are combined using SRSS, CQC and other
responses are combined using SRSS, CQC and other methods available in STAAD.methods available in STAAD.
The command syntax for defining response spectrum data is explained in Section The command syntax for defining response spectrum data is explained in Section 5.32.10
5.32.10.1 of the .1 of the Technical Reference manual.Technical Reference manual.
It is important to understand that once the combination methods like SRSS or CQC It is important to understand that once the combination methods like SRSS or CQC are applied, the sign of the results is lost. Consequently, results of a spectrum are applied, the sign of the results is lost. Consequently, results of a spectrum analysis, like displacements, forces and reactions do not have any
analysis, like displacements, forces and reactions do not have any sign.sign.
Because spectrum analysis requires modes and frequencies, the mass data and other Because spectrum analysis requires modes and frequencies, the mass data and other details explained in the chapter on calculating modes and frequencies are all details explained in the chapter on calculating modes and frequencies are all applicable in the case of spectrum analysis also. In other words, the mode and applicable in the case of spectrum analysis also. In other words, the mode and frequency calculation is a pre-requisite to
frequency calculation is a pre-requisite to performing responperforming response spectrum analysis.se spectrum analysis.
Calculation of Base Shear in a Response Spectrum Analysis Calculation of Base Shear in a Response Spectrum Analysis
The base shear, for a
The base shear, for a given mode for a given direction, reported in the responsegiven mode for a given direction, reported in the response spectrum analysis is obtained as
spectrum analysis is obtained as A * B * C * D
A * B * C * D where
where A = Mass
A = Mass participation factor for that mode for that directionparticipation factor for that mode for that direction B = Total mass specified for that direction
B = Total mass specified for that direction C = Spectral acceleration
C = Spectral acceleration for that modefor that mode
D = direction factor specified in that load case D = direction factor specified in that load case A is calculated by
A is calculated by the program from the mass matrix and mode shapesthe program from the mass matrix and mode shapes B is obtained from the masses
B is obtained from the masses specified in the response spectrum load casespecified in the response spectrum load case
C is obtained by interpolating between the user provided values of period vs.
C is obtained by interpolating between the user provided values of period vs.
acceleration and multiplying the resulting value
acceleration and multiplying the resulting value by the SCALE FACTOR.by the SCALE FACTOR.
D
D is specified by the user D is specified by the user
Bending Moment Diagram for a load case that involves the Response Spectrum Analysis Bending Moment Diagram for a load case that involves the Response Spectrum Analysis
In a response spectrum analysis in STAAD, the member forces are computed modes does not allow for the determination of the sign of the forces. Further, these modes does not allow for the determination of the sign of the forces. Further, these force values do not necessarily indicate whether these forces occur at the same force values do not necessarily indicate whether these forces occur at the same instant of time.
instant of time.
In order to
In order to draw the bending moment diagram, one needs to know the draw the bending moment diagram, one needs to know the moments at themoments at the intermediate section points on the member. In order to calculate these section force intermediate section points on the member. In order to calculate these section force values, the forces at the member ends have to be used. However, due to the special values, the forces at the member ends have to be used. However, due to the special nature of these end force values as described in the paragraph above, it makes no nature of these end force values as described in the paragraph above, it makes no sense to calculate the intermediate section forces based on the end force values.
sense to calculate the intermediate section forces based on the end force values.
Due to this reasoning, the bending moment diagram simply cannot be drawn Due to this reasoning, the bending moment diagram simply cannot be drawn accurately for the response spectrum loading. STAAD merely plots a straight line accurately for the response spectrum loading. STAAD merely plots a straight line that joins the bending moment values at
that joins the bending moment values at the start and the start and end joints of the member whichend joints of the member which are as mentioned earlier, absolute (positive) values. Current versions of STAAD do are as mentioned earlier, absolute (positive) values. Current versions of STAAD do not let the user draw the diagram at all from certain places such as the Member results of an equivalent UBC static analysis
results of an equivalent UBC static analysis
For the following reasons, this
For the following reasons, this comparison isn't meaningful :comparison isn't meaningful :
1.
1. In a spectrum analysis, In a spectrum analysis, the number of modes to be the number of modes to be combined is a decision madecombined is a decision made by
by ththe e enengigineneerer. . If If 10100% 0% papartrticicipipatatioion n frfrom om ththe e momodedes s isisn'n't t ututililizized ed in in ththee displacement calculation, it is obvious that the results
displacement calculation, it is obvious that the results will be only approximate.will be only approximate.
2.
2. In a spectrum analysis, the contribution from the various modes is combinedIn a spectrum analysis, the contribution from the various modes is combined using an SRSS method or a CQC method, both of which are only approximate using an SRSS method or a CQC method, both of which are only approximate methods. One very important drawback of both these methods is that
methods. One very important drawback of both these methods is that the sign ofthe sign of the displacements and forces cannot be determined. Also, the results can vary the displacements and forces cannot be determined. Also, the results can vary significantly depending on the type of
significantly depending on the type of method used in the method used in the combinationcombination..
3.
3. In the UBC method, only a single period is used. Normally, the assumption isIn the UBC method, only a single period is used. Normally, the assumption is that this period is associated
that this period is associated with a mode that encompasses a significant portionwith a mode that encompasses a significant portion of the overall response of the structure. This may not necessarily be true in of the overall response of the structure. This may not necessarily be true in reality. If more than one mode is required to capture the overall response of the reality. If more than one mode is required to capture the overall response of the structure, that fact is
structure, that fact is not brought to light in the UBC static not brought to light in the UBC static equivalent approach.equivalent approach.
4.
4. The UBC static equivalent method involves several parameters such asThe UBC static equivalent method involves several parameters such as Importance factor, soil structure
Importance factor, soil structure coefficient, etc. which are coefficient, etc. which are incorporated throughincorporated through an emperical formula. In a response spectrum analysis, there is no facility an emperical formula. In a response spectrum analysis, there is no facility
D D O O N N O O T T D D I I S S T T R R I I B B U U T T E E - - P P r r i i n n t t i i n n g f g f o o r r S S t t u u d d e e n n t t U U s s e e i i s s P P e e r r m m i i t t t t e e d d S S t t u d u d e e n n t t W W i i r r a a H H e e r r u u c c a a : : k k r r a a C C o o m m p p a a n n y y : : P P T T D D i i n n a a m m i k i k a a T T e e k k n n i i k k P P e e r r s s a a d d a a C l C l a a s s s s D D a a t t e e 0 0 9 9 O O c c t t 2 2 0 0 1 1 2 2 : : - - -
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Due to these reasons, a direct comparison of the results of a spectrum analysis and a Due to these reasons, a direct comparison of the results of a spectrum analysis and a static equivalent approach is
static equivalent approach is not recommended.not recommended.
Question :
Question : What What is the is the Scale FactoScale Factor (f4) r (f4) that needthat needs to s to be probe provided vided whenwhen specifying the Response Spectra?
specifying the Response Spectra?
Answer :
Answer : The The spectrum spectrum data data consists consists of of pairs pairs of of values values which which are are Period Period vs.vs.
Accn. or Period vs. Displacement. The acceleration or displacement values that you Accn. or Period vs. Displacement. The acceleration or displacement values that you obtain from the geological data for that site may have been provided to you as obtain from the geological data for that site may have been provided to you as normalized values or un-normalized values. Normalization means that the values of normalized values or un-normalized values. Normalization means that the values of acceleration or displacement have been divided by a number (called normalization acceleration or displacement have been divided by a number (called normalization factor) which represents some reference value. One of the commonly used factor) which represents some reference value. One of the commonly used normalization factors is 'g', the acceleration due to
normalization factors is 'g', the acceleration due to gravity.gravity.
If the spectrum data you specify in STAAD is a normalized spectrum data, you If the spectrum data you specify in STAAD is a normalized spectrum data, you should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your spectrum data is un-normalized, there is no need to provide a scale factor(Another spectrum data is un-normalized, there is no need to provide a scale factor(Another way of putting it is that if you provide un-normalized spectrum values, the scale way of putting it is that if you provide un-normalized spectrum values, the scale factor is 1, which happens to be the default value also.) Make sure
factor is 1, which happens to be the default value also.) Make sure that the value youthat the value you pr
provovidide e fofor r ththe e SCSCALALE E FAFACTCTOR OR is is in in acaccocordrdanance ce wiwith th ththe e lelengngth th ununitits s yoyou u hahaveve specified. (A common error is that if the
specified. (A common error is that if the scale factor is scale factor is 'g', users erroneously provide'g', users erroneously provide 32.2 when the length unit is in INCHES.)
32.2 when the length unit is in INCHES.)
STAAD will multiply the spectral acceleration or
STAAD will multiply the spectral acceleration or spectral displacement values by thespectral displacement values by the scale factor. Hence, if you provide a normalized acceleration value of 0.5 and a scale scale factor. Hence, if you provide a normalized acceleration value of 0.5 and a scale factor of 386.4 inch/sq.sec., it has the same effect as providing an un-normalized factor of 386.4 inch/sq.sec., it has the same effect as providing an un-normalized acceleration value of 193.2 inch/sq.sec. and a scale
acceleration value of 193.2 inch/sq.sec. and a scale factor of 1.0.factor of 1.0.
Question :
Question : What is the Direction Factor thWhat is the Direction Factor that needs to be provided at needs to be provided when specifyinwhen specifyingg the Response Spectra?
the Response Spectra?
Answer :
Answer : The The Direction Direction factor factor is is a a quantity quantity by by which which the the spectral spectral displacementdisplacement for the associated direction is
for the associated direction is multiplied.multiplied.
For example, if the command reads as For example, if the command reads as
SPECTRUM SRSS X 0.7 Y 0.5 Z 0.65 DISP DAMP 0.05 SCALE 32.2 SPECTRUM SRSS X 0.7 Y 0.5 Z 0.65 DISP DAMP 0.05 SCALE 32.2 the following is done:
the following is done:
1.
1. For each mode, the period is determined.For each mode, the period is determined.
2.
2. Corresponding to the period, the spectral displacement for that mode isCorresponding to the period, the spectral displacement for that mode is calculated by interpolation from the input pairs of period vs. spectral calculated by interpolation from the input pairs of period vs. spectral displacement. Call this "sd"
displacement. Call this "sd"
D D O O N N O O T T D D I I S S T T R R I I B B U U T T E E - - P P r r i i n n t t i i n n g f g f o o r r S S t t u u d d e e n n t t U U s s e e i i s s P P e e r r m m i i t t t t e e d d S S t t u d u d e e n n t t W W i i r r a a H H e e r r u u c c a a : : k k r r a a C C o o m m p p a a n n y y : : P P T T D D i i n n a a m m i k i k a a T T e e k k n n i i k k P P e e r r s s a a d d a a C l C l a a s s s s D D a a t t e e 0 0 9 9 O O c c t t 2 2 0 0 1 1 2 2 : : - - -
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3.
3. Calculate the spectral displacement for each direction by multiplying "sd" byCalculate the spectral displacement for each direction by multiplying "sd" by the associated Direction factor.
the associated Direction factor.
The X direction spectral displacement = sd
The X direction spectral displacement = sd * 0.7* 0.7 The Y direction spectral displacement = sd
The Y direction spectral displacement = sd * 0.5* 0.5 The Z direction spectral displacement = sd
The Z direction spectral displacement = sd * 0.65* 0.65 These factored values are then multiplied by
These factored values are then multiplied by
a.
a. the mode shape value the mode shape value corresponding to that degree of corresponding to that degree of freedom,freedom,
b.
b. pa partrticicipipatatioion n fafactctoror..
Call the result T(m) where
Call the result T(m) where "m" stands for the mode number."m" stands for the mode number.
Once the T(m) is determined for all modes, subject them to the SRSS calculation.
Once the T(m) is determined for all modes, subject them to the SRSS calculation.
That will provide the
That will provide the node displacement corresponding to that degree of node displacement corresponding to that degree of freedom.freedom.
Question :
Question : The results The results of the of the response spectruresponse spectrum load m load case are alwcase are always positivays positivee numbers. Why? How do I know that the positive value is always critical, especially numbers. Why? How do I know that the positive value is always critical, especially from the design standpoint?
from the design standpoint?
Answer :
Answer : In In a a spectrum spectrum analysis, analysis, the the contribution contribution of of the the individual individual modes modes isis combined using method
combined using methods such as s such as SRSS or CQC to arrive at SRSS or CQC to arrive at the overall response. Thethe overall response. The limitation of these methods is that the sign of the response cannot be determined limitation of these methods is that the sign of the response cannot be determined after the method is applied. This is the reason why the output you get from STAAD after the method is applied. This is the reason why the output you get from STAAD for a response spectrum analysis are
for a response spectrum analysis are absolute values.absolute values.
One way to deal with the problem is to create 2 load combination cases for each set One way to deal with the problem is to create 2 load combination cases for each set of load cases you wish to combine. For example, if the dead load case is 1, and the of load cases you wish to combine. For example, if the dead load case is 1, and the spectrum load case is 5,
spectrum load case is 5, you could createyou could create LOAD COMB 10
LOAD COMB 10 1
1 1.1 1.1 5 5 1.31.3
LOAD COMB 11 LOAD COMB 11 1 1.1 5 -1.3
1 1.1 5 -1.3
and use the critical value from amongst these 2 load combination cases for design and use the critical value from amongst these 2 load combination cases for design pu
purprpososeses. . WhWhat at yoyou u acaccocompmplilish sh frfrom om ththis is prprococesess s is is ththat at yoyou u arare e coconsnsididererining g aa po
posisititive ve efeffefect ct as as wewell ll as as ththe e nenegagatitive ve efeffefect ct of of ththe e spspecectrtrum um loload ad cacasese..
Question :
Question : In the Technical RIn the Technical Reference manual section 5eference manual section 5.32.10..32.10.1, you state: " No1, you state: " Note,te, if data is in g
if data is in g acceleration units, then set SCALE to a conversion factor to the currentacceleration units, then set SCALE to a conversion factor to the current length unit (9.81, 386.4, etc.)"
length unit (9.81, 386.4, etc.)"
What does "g acceleration units" mean?
What does "g acceleration units" mean?
Related question :
Related question : What is the Scale Factor (f4) that needs to be What is the Scale Factor (f4) that needs to be pr
provovidided ed whwhen en spspececififyiying ng ththe e ReRespspononse se SpSpecectrtra?a?
D D O O N N O O T T D D I I S S T T R R I I B B U U T T E E - - P P r r i i n n t t i i n n g f g f o o r r S S t t u u d d e e n n t t U U s s e e i i s s P P e e r r m m i i t t t t e e d d S S t t u d u d e e n n t t W W i i r r a a H H e e r r u u c c a a : : k k r r a a C C o o m m p p a a n n y y : : P P T T D D i i n n a a m m i k i k a a T T e e k k n n i i k k P P e e r r s s a a d d a a C l C l a a s s s s D D a a t t e e 0 0 9 9 O O c c t t 2 2 0 0 1 1 2 2 : : - - -
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Answer :
Answer : The The spectrum spectrum data data consists consists of of pairs pairs of of values values which which are are Period Period vs.vs.
Accn. or Period vs. Displacement. The acceleration or displacement values that you Accn. or Period vs. Displacement. The acceleration or displacement values that you obtain from the geological data for that site may have been provided to you as obtain from the geological data for that site may have been provided to you as normalized values or un-normalized values. Normalization means that the values of normalized values or un-normalized values. Normalization means that the values of acceleration or displacement have been divided by a number (called normalization acceleration or displacement have been divided by a number (called normalization factor) which represents some reference value. One of the commonly used factor) which represents some reference value. One of the commonly used normalization factors is 'g', the acceleration due to
normalization factors is 'g', the acceleration due to gravity.gravity.
If the spectrum data you specify in STAAD is a normalized spectrum data, you If the spectrum data you specify in STAAD is a normalized spectrum data, you should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your spectrum data is un-normalized, there is no need to provide a scale factor(Another spectrum data is un-normalized, there is no need to provide a scale factor(Another
If the spectrum data you specify in STAAD is a normalized spectrum data, you If the spectrum data you specify in STAAD is a normalized spectrum data, you should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your should provide the NORMALIZATION FACTOR as the SCALE FACTOR. If your spectrum data is un-normalized, there is no need to provide a scale factor(Another spectrum data is un-normalized, there is no need to provide a scale factor(Another