TR.32.10.1.2 Response Spectrum Specification per NRC 2005

This command may be used to specify and apply the RESPONSE SPECTRUM loading as per the 2005 edition of the National Research Council specification National Building Code of Canada (NBC) , for dynamic analysis. The graph of frequency – acceleration pairs are calculated based on the input requirements of the command and as defined in the code.

General Format

SPECTRUM comb-method NRC 2005 ( TORSION (DECCENTRICITY f20) (ECCENTRICITY f21) ) *{ X f1 | Y f2| Z f3} { ACC | DIS } ( SCALE f4)

{ DAMP f5| CDAMP | MDAMP } ( { LINEAR | LOGARITHMIC } ) (MISSING f6) (ZPA f7) ({ DOMINANT f10| SIGN }) (SAVE)  (IMR f11) (STARTCASE f12)

Note: The data from SPECTRUM through SCALE must be on the first line of the command. The data shown on the second line above can be continued on the first line or one or more new lines with all but last ending with a hyphen (limit of four lines per spectrum).

The command is completed with the following spectrum data which must be started on a new line:

{ p1 v1; p2 v2; p3 v3; … pn vn | FILE filename  }

Where:

Table 1. Parameters used for NRC 2005 response spectrum
Parameter	Default Value	Description
DECCENTRICITY f20	1.0	Factor to be multiplied with static eccentricity (i.e., eccentricity between center of mass and center of rigidity).
ECCENTRICITY f21	0.05	Factor for accidental eccentricity. Positive values indicate clockwise torsion and negative values indicate counterclockwise torsion.
X f1, Y f2, Z f3	0.0	Factors for the input spectrum to be applied in X, Y, & Z directions. Any one or all directions can be input. Directions not provided will default to zero.
SCALE f4	1.0	Linear scale factor by which the spectra data will be multiplied. Usually used to factor g’s to length/sec² units. This input is the appropriate value of acceleration due to gravity in the current unit system (thus, 9.81 m/s² or 32.2 ft/s²).
DAMP f5	0.05	The damping ratio. Specify a value of exactly 0.0000011 to ignore damping. If `CDAMP` is specified, then composite damping is used as determined by the values for material damping (and spring damping, if specified). Refer to TR.26.2 Specifying Constants for Members and Elements If `MDAMP` is specified, then modal damping is calculated using the method defined in a `DEFINE DAMPING INFORMATION` command, which must be included in the input file. Refer to TR.26.4 Modal Damping Information
MISSING f6		Optional parameter to use "Missing Mass" method. The static effect of the masses not represented in the modes is included. The spectral acceleration for this missing mass mode is the f6value entered in length/sec² (this value is not multiplied by SCALE). If f6is zero, then the spectral acceleration at the `ZPA` f7frequency is used. If f7is zero or not entered, the spectral acceleration at 33Hz (Zero Period Acceleration, ZPA) is used. The results of this calculation are SRSSed with the modal combination results. Note: If the `MISSING` parameter is entered on any spectrum case it will be used for all spectrum cases.
ZPA f7	33 [Hz]	The zero period acceleration value for use with `MISSING` option only. Defaults to 33 Hz if not entered. The value is printed but not used if `MISSING f6` is entered.
DOMINANT f10	1 (1st Mode)	The dominant mode method. All results will have the same sign as mode number f10 alone would have if it were excited then the scaled results were used as a static displacements result. Defaults to mode 1 if no value entered. If a 0 value entered, then the mode with the greatest % participation in the excitation direction will be used (only one direction factor may be nonzero). The dominant mode is selected based on the actual base shear of the mode and not the greatest % participation factor. Note: Do not enter the `SIGN` parameter with this option. Ignored for the `ABS` method of combining spectral responses from each mode.
IMR f11	1	The number of individual modal responses (scaled modes) to be copied into load cases. Defaults to one. If greater than the actual number of modes extracted (`NM`), then it will be reset to NM. Modes one through f11 will be used. Missing Mass modes are not output.
STARTCASE f12	Highest Load Case No. + 1	The primary load case number of mode 1 in the `IMR` parameter. Defaults to the highest load case number used so far plus one. If f12 is not higher than all prior load case numbers, then the default will be used. For modes 2 through `NM`, the load case number is the prior case number plus one.

comb-method = { SRSS | ABS | CQC | ASCE | TEN | CSM | GRP } are methods of combining the responses from each mode into a total response.

The CQC and ASCE4-98 methods require damping. ABS, SRSS, CRM, GRP, and TEN methods do not use damping unless spectra-period curves are made a function of damping (see File option below). CQC, ASCE, CRM, GRP, and TEN include the effect of response magnification due to closely spaced modal frequencies. ASCE includes more algebraic summation of higher modes. ASCE and CQC are more sophisticated and realistic methods and are recommended.

SRSS

Square Root of Summation of Squares method.

ABS

Absolute sum. This method is very conservative and represents a worst case combination.

CQC

Complete Quadratic Combination method (Default). This method is recommended for closely spaced modes instead of SRSS.

Resultants are calculated as:

F = \sqrt{\sum_{n} \sum_{m} f_{n} ρ_{n m} f_{m}}

where

ρ_nm	=	$\frac{8 ζ^{2} (1 + r) r^{2 / 3}}{{(1 - r^{2})}^{2} + 4 ζ^{2} r {(1 + r)}^{2}}$
r	=	ω_n/ω_m ≤ 1.0

Note: The cross-modal coefficient array is symmetric and all terms are positive.

ASCE

NRC Regulatory Guide Rev. 2 (2006) Gupta method for modal combinations and Rigid/Periodic parts of modes are used. The ASCE4-98 definitions are used where there is no conflict. ASCE4-98 Eq. 3.2-21 (modified Rosenblueth) is used for close mode interaction of the damped periodic portion of the modes.

TEN

Ten Percent Method of combining closely spaced modes. NRC Reg. Guide 1.92 (Rev. 1.2.2, 1976).

CSM

Closely Spaced Method as per IS:1893 (Part 1)-2002 procedures.

GRP

Closely Spaced Modes Grouping Method. NRC Reg. Guide 1.92 (Rev. 1.2.1, 1976).

Note: If SRSS is selected, the program will internally check whether there are any closely spaced modes or not. If it finds any such modes, it will switch over to the CSM method. In the CSM method, the program will check whether all modes are closely spaced or not. If all modes are closely spaced, it will switch over to the CQC method.

TORSION

indicates that the torsional moment (in the horizontal plane) arising due to eccentricity between the center of mass and center of rigidity needs to be considered. See Inherent and Accidental Torsion for additional information.

Note: If TORSION is entered on any one spectrum case it will be used for all spectrum cases.

Lateral shears at story levels are calculated in global X and Z directions. For global Y direction the effect of torsion will not be considered.

ACCELERATION or DISPLACEMENT

indicates whether Acceleration or Displacement spectra will be entered. The relationship between acceleration and displacement values in response spectra data is:

Displacement = Acceleration \times {(1 / ω)}^{2}

where

2π/Period (period given in seconds; ω in cycles per second)

DAMP, MDAMP, and CDAMP

select source of damping input:

DAMP indicates to use the f2 value for all modes
MDAMP indicates to use the damping entered or computed with the DEFINE DAMP command if entered, otherwise default value of 0.05 will be used
CDAMP indicates to use the composite damping of the structure calculated for each mode. You must specify damping for different materials under the CONSTANT specification

LINEAR or LOGARITHMIC

Select Linear or Logarithmic interpolation of the input Spectra versus Period curves for determining the spectra value for a mode given its period. Linear is the default. Since Spectra versus Period curves are often linear only on Log-Log scales, the logarithmic interpolation is recommended in such cases; especially if only a few points are entered in the spectra curve.

When FILE filename is entered, the interpolation along the damping axis will be linear.

Note: The last interpolation parameter entered on the last of all of the spectrum cases will be used for all spectrum cases.

SIGN

This option results in the creation of signed values for all results. The sum of squares of positive values from the modes are compared to sum of squares of negative values from the modes. If the negative values are larger, the result is given a negative sign. This command is ignored for ABS option.

CAUTION: Do not enter DOMINANT parameter with this option.

SAVE

This option results in the creation of a acceleration data file (with the model file name and an .acc file extension) containing the joint accelerations in g’s and radians/sec². These files are plain text and may be opened and viewed with any text editor (e.g., Notepad).

Spectra data is input in one of these two input forms:

p1, v1; p2, v2; …. ; pn, vn. Data is part of input, immediately following the SPECTRUM command. Period – Value pairs (separated by semi colons) are entered to describe the Spectrum curve. Period is in seconds and the corresponding Value is either acceleration (current length unit/sec²) or displacement (current length unit) depending on the ACC or DIS chosen. Continue the curve data onto as many lines as needed (up to 500 spectrum pairs). Spectrum pairs must be in ascending order of period. Note, if data is in g acceleration units, then set SCALE to a conversion factor to the current length unit (9.81, 386.4, etc.). Also note, do not end these lines with a hyphen (-). Each SPECTRUM command must be followed by Spectra data if this input form is used.
FILE filename data is in a separate file, using the format described in File Format for Spectra Data.

When the File filename command has been provided, then you must have the spectra curve data on a file named filename prior to starting the analysis. The format of the FILE spectra data allows spectra as a function of damping as well as period.

Note: If the FILE filename command is entered, it must be with the first spectrum case and will be used for all spectrum cases.

No File filename command needs to be entered with the remaining spectrum cases. The filename may not be more than 72 characters in length.

Individual Modal Response Case Generation

Individual modal response (IMR) cases are simply the mode shape scaled to the magnitude that the mode has in this spectrum analysis case before it is combined with other modes. If the IMR parameter is entered, then STAAD.Pro will create load cases for the first specified number of modes for this response spectrum case (i.e., if five is specified then five load cases are generated, one for each of the first five modes). Each case will be created in a form like any other primary load case.

The results from an IMR case can be viewed graphically or through the print facilities. Each mode can therefore be assessed as to its significance to the results in various portions of the structure. Perhaps one or two modes could be used to design one area/floor and others elsewhere.

You can use subsequent load cases with TR.32.11 Repeat Load Specification combinations of these scaled modes and the static live and dead loads to form results that are all with internally consistent signs (unlike the usual response spectrum solutions). The modal applied loads vector will be omega squared times mass times the scaled mode shape. Reactions will be applied loads minus stiffness matrix times the scaled mode shape.

With the Repeat Load capability, you can combine the modal applied loads vector with the static loadings and solve statically with P-Delta or tension only.

Note: When the IMR option is entered for a Spectrum case, then a TR.37 Analysis Specification & TR.38 Change Specification must be entered after each such Spectrum case.

TR.32.10.1.1 Response Spectrum Specification - Custom for additional details on IMR load case generation.

Inherent and Accidental Torsion

In response spectrum analysis all the response quantities (i.e., joint displacements, member forces, support reactions, plate stresses, etc.) are calculated for each mode of vibration considered in the analysis. These response quantities from each mode are combined using a modal combination method (either by CQC, SRSS, ABS, TEN PERCENT, etc.) to produce a single positive result for the given direction of acceleration. This computed result represents a maximum magnitude of the response quantity that is likely to occur during seismic loading. The actual response is expected to vary from a range of negative to positive value of this maximum computed quantity.

No information is available from response spectrum analysis as to when this maximum value occurs during the seismic loading and what will be the value of other response quantities at that time. As for example, consider two joints J2 and J3 whose maximum joint displacement in global X direction come out to be X1 and X2 respectively. This implies that during seismic loading joint J1 will have X direction displacement that is expected to vary from -X1 to +X1 and that for joint J2 from -X2 to +X2. However, this does not necessarily mean that the point of time at which the X displacement of joint J1 is X1, the X displacement of joint J2 will also be X2.

For the reason stated above, torsional moment at each floor arising due to dynamic eccentricity along with accidental eccentricity (if any) is calculated for each mode. Lateral story shear from this torsion is calculated forming global load vectors for each mode. Static analysis is carried out with this global load vector to produce global joint displacement vectors for each mode due to torsion. These joint displacements from torsion for each mode are algebraically added to the global joint displacement vectors from response spectrum analysis for each mode. The final joint displacements from response spectrum along with torsion for all modes are combined using specified modal combination method to get final maximum possible joint displacements. Refer to the steps explained below.

Steps

For each mode following steps are performed to include Torsion provision.

Lateral story force at each floor is calculated.
At each floor design eccentricity is calculated.

Thus, design eccentricity e_di = f20×e_si + f12×b_i where f20 = 1.0 and f21 = (±) 0.05
where
e_si
=
static eccentricity between center of mass and center of rigidity at floor i
b_i
=
floor plan dimension in the direction of earthquake loading
Torsional moment is calculated at each floor.

M_ik = Q_ik × e_di at floor i for mode k
The lateral nodal forces corresponding to torsional moment are calculated at each floor. These forces represent the additional story shear due to torsion.
Static analysis of the structure is performed with these nodal forces.
The analysis results (i.e., member force, joint displacement, support reaction, etc) from torsion are algebraically added to the corresponding modal response quantities from response spectrum analysis.
Steps 1 through 6 are performed for all modes considered and missing mass correction (if any). Finally, the peak response quantities from the different modal responses are combined as per the specified combination method (e.g., SRSS, CQC, TEN, etc.)

Dynamic Eccentricity

The static eccentricity is generally defined as the distance between the center of mass (CM) and the center of rigidity (CR) at respective floors levels. Accidental eccentricity generally accounts for factors such as:

the rotational component of ground motion about the vertical axis,
the difference between computed and actual values of the mass, stiffness, or strength, and
uneven live mass distribution.

The provision for design eccentricity e_di at i^th floor of a building is given by the following equation:

e_di = DEC×e_si + ECC×b_i

where

e_si	=	static eccentricity at i^th floor
b_i	=	plan dimension of the i^th floor normal to the direction of ground motion
`ECC` and `DEC`	=	Factors to determine the design eccentricity. These are input parameters.

Example

LOAD 1 LOADTYPE None  TITLE RS_X
SPECTRUM SRSS NRC 2005 X 0.5 ACC DAMP 0.05 LIN
0 9.80665; 0.06 18.6326; 0.12 24.5166; 0.18 24.5166; 0.24 24.5166; 0.3 24.5166;
0.36 24.5166; 0.42 23.3492; 0.48 20.4305; 0.54 18.1605; 0.6 16.3444;
0.66 14.8586; 0.72 13.6203; 0.78 12.5726; 0.84 11.6746; 0.9 10.8963;
0.96 10.2153; 1.02 9.61436; 1.08 9.08023; 1.14 8.60233; 1.2 8.17221;
1.26 7.78306; 1.32 7.42928; 1.38 7.10627; 1.44 6.81018; 1.5 6.53777;
1.56 6.28632; 1.62 6.05349; 1.68 5.83729; 1.74 5.63601; 1.8 5.44814;
1.86 5.2724; 1.92 5.10763; 1.98 4.95286; 2.04 4.80718; 2.1 4.66984;
2.16 4.54012; 2.22 4.41741; 2.28 4.30116; 2.34 4.19088; 2.4 4.08611;
2.46 3.98645; 2.52 3.89153; 2.58 3.80103; 2.64 3.71464; 2.7 3.63209;
2.76 3.55314; 2.82 3.47754; 2.88 3.40509; 2.94 3.3356; 3 3.26889; 3.06 3.20479;
3.12 3.14316; 3.18 3.08385; 3.24 3.02675; 3.3 2.97171; 3.36 2.91865;
3.42 2.86744; 3.48 2.818; 3.54 2.77024; 3.6 2.72407; 3.66 2.67941; 3.72 2.6362;
3.78 2.59435; 3.84 2.55382; 3.9 2.51453; 3.96 2.47643; 4.02 2.45166;
4.08 2.45166; 4.14 2.45166; 4.2 2.45166; 4.26 2.45166; 4.32 2.45166;
4.38 2.45166; 4.44 2.45166; 4.5 2.45166; 4.56 2.45166; 4.62 2.45166;
4.68 2.45166; 4.74 2.45166; 4.8 2.45166; 4.86 2.45166; 4.92 2.45166;
4.98 2.45166; 5.04 2.45166; 5.1 2.45166; 5.16 2.45166; 5.22 2.45166;
5.28 2.45166; 5.34 2.45166; 5.4 2.45166; 5.46 2.45166; 5.52 2.45166;
5.58 2.45166; 5.64 2.45166; 5.7 2.45166; 5.76 2.45166; 5.82 2.45166;
5.88 2.45166; 5.94 2.45166;

e_si	=	static eccentricity between center of mass and center of rigidity at floor i
b_i	=	floor plan dimension in the direction of earthquake loading