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V. NRC 2010 Response Spectrum

Calculation of Base Shear for Response Spectrum Method in NBCC/NRC 2010.

Details

Example 4.7(1) is modeled in STAAD.Pro.

The generated model is subjected to a Response Spectrum Load along global X direction. Time-Acceleration pairs for the spectrum data can be seen in supporting excel file NRC_Calculation.xlsx within worksheet named RS Values. The direction factor along global X direction is assumed as 0.5. The base shear reported by STAAD.Pro is verified against hand calculation.

Validation

Natural Frequency/Time Period Information & Mode Shapes (Obtained from output file)

Table 1. Natural Frequencies/Time Period
Mode Frequency (cyc/sec) Time Period (sec)
1 3.332 0.30014
2 9.103 0.10986
3 12.435 0.08042
Table 2. Mode Shapes
Mode Story Level/Joint Numbers X-Trans Y-Trans Z-Trans X-Rot Y-Rot Z-Rot
1 Roof/Joints 7 & 8 1 0 0 0 0 0
2nd Floor/Joints 5 & 6 0.86603 0 0 0 0 0
1st Floor/Joints 2 & 3 0.5 0 0 0 0 0
2 Roof/Joints 7 & 8 -1 0 0 0 0 0
2nd Floor/Joints 5 & 6 0 0 0 0 0 0
1st Floor/Joints 2 & 3 1 0 0 0 0 0
3 Roof/Joints 7 & 8 1 0 0 0 0 0
2nd Floor/Joints 5 & 6 -0.86603 0 0 0 0 0
1st Floor/Joints 2 & 3 0.5 0 0 0 0 0
Table 3. Mode Participation Factor Calculation
Story Level Weight Wi (KN) Mode 1 Mode 2 Mode 3
ϕ Wi x ϕ Wi x ϕ 2 ϕ Wi x ϕ Wi x ϕ 2 ϕ Wi x ϕ Wi x ϕ 2
Roof 49.035 1 49.035 49.035 -1 -49.035 49.035 1 49.035 49.035
2nd Floor 98.07 0.86603 84.932 73.553 0 0 0 -0.86603 -84.932 73.553
1st Floor 98.07 0.5 49.035 24.518 1 98.07 98.07 0.5 49.035 24.518
Summation 245.175 183.00 147.11 49.035 147.11 13.138 147.11
Modal Weight Mk x g 227.7 16.345 1.173
Modal Weight Participation (In %) 92.85 6.667 0.4786
Mode Participation Factor ∏ ∑Wi x ϕ / ∑ Wi x ϕ 2 1.244 0.3333 0.0893
Table 4. Horizontal Acceleration Spectrum Value Calculation
Mode 1 Mode 2 Mode 3
Time Period T 0.30014 0.10986 0.08042
Spectrum Data Obtain from NRC_Calculation.xlsx within worksheet named RS Values
Formula Linearly Interpolate acceleration values between 0.3 sec & 0.36 sec to obtain acceleration value for 0.30014 sec. Linearly Interpolate acceleration values between 0.06 sec & 0.12 sec to obtain acceleration value for 0.10986 sec. Linearly Interpolate acceleration values between 0.06 sec & 0.12 sec to obtain acceleration value for 0.08042 sec.
Sa 1.50 1.40 1.10
Horizontal Seismic Coefficient Cs = (Sa x Direction Factor) = Sa x 0.5
  1. 249998725
1.25
1.20 1.05
Table 5. Base Shear Calculation and its distribution across height
Story Level Weight Wi (KN) Mode 1 Mode 2 Mode 3 Story Shear due to all modes i.e. SRSS of all modes
ϕ Qi (Cs x ϕ x ∏ x Wi) Vi (∑Qi) ϕ Qi (Cs x ϕ x ∏ x Wi) Vi (∑Qi) ϕ Qi (Cs x ϕ x ∏ x Wi) Vi (∑Qi)
Roof 49.035 1 76.25 76.25 -1 -19.60 -19.60 1 4.608 4.608 78.86
2nd Floor 98.07 0.86603 132.1 208.3 0 0 -19.60 -0.86603 -7.981 -3.373 209.27
1st Floor 98.07 0.5 76.25 284.6 1 39.21 19.60 0.5 4.608 1.235 285.25
Base Shear V 285.25

Comparison

Table 6. Comparison of results
Parameter STAAD.Pro Reference Difference Comments
Base shear (kN) 285.25 285.25 none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\06 Loading\NRC\2010\NRC 2010 Response Spectrum.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 08-Feb-17
END JOB INFORMATION
INPUT WIDTH 79
SET SHEAR
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 3 0; 3 3 3 0; 4 3 0 0; 5 0 6 0; 6 3 6 0; 7 0 9 0; 8 3 9 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 2 5; 5 3 6; 6 5 6; 7 5 7; 8 6 8; 9 7 8;
START USER TABLE
TABLE 1
UNIT METER KN
PRISMATIC
COLLUMN
1e+09 0.000847246 0.001 0.001 0.001 0.001 0.5 0.5
TABLE 2
UNIT METER KN
PRISMATIC
BEAM
0.001 1e+09 0.001 0.001 0.001 0.001 0.5 0.5
END
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 2.17185e+07
POISSON 0.17
DENSITY 23.5616
ALPHA 1e-05
DAMP 0.05
TYPE CONCRETE
STRENGTH FCU 27579
G 9.28139e+06
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 3 TO 5 7 8 UPTABLE 1 COLLUMN
2 6 9 UPTABLE 2 BEAM
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 4 FIXED
DEFINE REFERENCE LOADS
LOAD R1 LOADTYPE Mass  TITLE REF LOAD CASE 1
JOINT LOAD
8 FX 49.035
3 6 FX 98.07
END DEFINE REFERENCE LOADS
FLOOR DIAPHRAGM
DIA 1 TYPE RIG HEI 3
DIA 2 TYPE RIG HEI 6
DIA 3 TYPE RIG HEI 9
LOAD 1 LOADTYPE None  TITLE RS_X
SPECTRUM SRSS NRC 2010 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;
PERFORM ANALYSIS PRINT ALL
PRINT MODE SHAPES
PRINT ANALYSIS RESULTS
FINISH

STAAD Output

               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
         1                       3.332                  0.30014
         2                       9.103                  0.10986
         3                      12.435                  0.08042
  RESPONSE SPECTRUM LOAD     1 
     RESPONSE LOAD CASE      1
            MODAL WEIGHT (MODAL MASS TIMES g) IN KN           GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       2.276565E+02  0.000000E+00  0.000000E+00    1.471050E+02
         2       1.634500E+01  0.000000E+00  0.000000E+00    1.471050E+02
         3       1.173519E+00  0.000000E+00  0.000000E+00    1.471051E+02
     SRSS          MODAL COMBINATION METHOD USED.
     DYNAMIC WEIGHT X Y Z   2.451750E+02  0.000000E+00  0.000000E+00 KN  
     MISSING WEIGHT X Y Z  -3.177132E-06  0.000000E+00  0.000000E+00 KN  
       MODAL WEIGHT X Y Z   2.451750E+02  0.000000E+00  0.000000E+00 KN  
           MODE                 ACCELERATION-G     DAMPING
           ----                 --------------     -------
              1                      2.50000       0.05000
              2                      2.39857       0.05000
              3                      2.10421       0.05000
      STAAD SPACE                                              -- PAGE NO.    8
 MODAL BASE ACTIONS 
  MODAL BASE ACTIONS        FORCES IN KN   LENGTH IN METE
  -----------------------------------------------------------
                                                             MOMENTS ARE ABOUT THE ORIGIN
   MODE     PERIOD        FX          FY          FZ          MX          MY          MZ
      1      0.300      284.57        0.00        0.00        0.00        0.00    -1707.42
      2      0.110       19.60        0.00        0.00        0.00        0.00       58.81
      3      0.080        1.23        0.00        0.00        0.00        0.00       -7.41
 PARTICIPATION FACTORS 
           MASS  PARTICIPATION FACTORS IN PERCENT         BASE SHEAR IN KN  
           --------------------------------------         ------------------
 MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z       X        Y        Z
   1   92.85  0.00  0.00   92.855    0.000    0.000    284.57     0.00     0.00
   2    6.67  0.00  0.00   99.521    0.000    0.000     19.60     0.00     0.00
   3    0.48  0.00  0.00  100.000    0.000    0.000      1.23     0.00     0.00
                                                    ---------------------------
                                  TOTAL SRSS  SHEAR    285.25     0.00     0.00
                                  TOTAL 10PCT SHEAR    285.25     0.00     0.00
                                  TOTAL ABS   SHEAR    305.41     0.00     0.00