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V. EN 1998-1-2004 Response Spectrum

Calculation of base shear for response spectrum method per EN 1998-1-2004.

Details

A three story building is modelled in STAAD.Pro considering the following:
  • Type 1
  • Ground type A
  • Soil factor, S = 1
  • Lower limit of period, TB = 0.15 sec
  • Upper limit of period, TC = 0.4 sec
  • Value defining the beginning of constant displacement response range, TD = 2 sec
  • Damping: 5%
  • Design ground acceleration, ag = 0.981 m/sec2
  • Damping correction factor, η = 1
  • Behavior factor, q = 3
  • Lower bound factor for horizontal design spectrum, β = 0.2
The generated model is subjected to a Response Spectrum Load along global X direction. 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 × ϕ Wi × ϕ 2 ϕ Wi × ϕ Wi × ϕ 2 ϕ Wi × ϕ Wi × 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 24.5175
Summation 245.175 183.00 147.11 49.035 147.105 13.138 147.106
Modal Weight Mk × g 227.66 16.35 1.173
Modal Weight Participation (In %) 92.86 6.667 0.479
Mode Participation Factor ∏ ∑Wi × ϕ / ∑ Wi × ϕ 2 1.244 0.333 0.089
Mode 1 Mode 2 Mode 3
Time Period T 0.30014 0.10986 0.08042
Design response spectrum Sd(T):

As TB≤ T≤ TC ,hence-

ag x S x (2.5/q)

= 0.981 x 1 x (2.5/3)

= 0.8175

As 0 ≤ T≤ TB ,hence-

ag x S x [(2/3) + (T/TB) x {(2.5/q)-(2/3)}]

= 0.981 x 1 x [(2/3) + (0.10986/0.15) x {(2.5/3)-(2/3)}]

= 0.7737

As 0 ≤ T≤ TB ,hence-

ag x S x [(2/3) + (T/TB) x {(2.5/q)-(2/3)}]

= 0.981 x 1 x [(2/3) + (0.08042/0.15) x {(2.5/3)-(2/3)}]

= 0.7417
Sd/g = 0.083 0.079 0.076

Design lateral force at each floor in each mode, Qi = Pk × si × (Sd/g) × Wi

Table 4. 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
si Qi Vi si Qi Vi si Qi Vi
Roof 49.035 1 5.08 5.08 -1 -1.29 -1.29 1 0.33 0.33 5.25
2nd Floor 98.070 0.86603 8.80 13.89 0 0.00 -1.29 -0.86603 -0.57 -0.24 13.95
1st Floor 98.070 0.5 5.08 18.97 1 2.58 1.29 0.5 0.33 0.09 19.02
Base Shear, V 19.02

Comparison

Table 5. Comparison of results
Parameter Reference STAAD.Pro Difference Comments
Base shear (kN) 19.02 19.02 none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\06 Loading\EN 1998-1-2004\EN 1998-1-2004 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 EURO 2004 DESIGN RS1 X 1 ACC DAMP 0.05
SOIL TYPE A ALPHA 0.1 Q 3
PERFORM ANALYSIS PRINT ALL
PRINT MODE SHAPES
PRINT ANALYSIS RESULTS
FINISH

STAAD Output

 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       18.97        0.00        0.00        0.00        0.00     -113.83
      2      0.110        1.29        0.00        0.00        0.00        0.00        3.87
      3      0.080        0.09        0.00        0.00        0.00        0.00       -0.53
 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     18.97     0.00     0.00
   2    6.67  0.00  0.00   99.521    0.000    0.000      1.29     0.00     0.00
   3    0.48  0.00  0.00  100.000    0.000    0.000      0.09     0.00     0.00
                                                    ---------------------------
                                  TOTAL SRSS  SHEAR     19.02     0.00     0.00
                                  TOTAL 10PCT SHEAR     19.02     0.00     0.00
                                  TOTAL ABS   SHEAR     20.35     0.00     0.00
Parent topic: V. EN 1998-1-2004