V. GB 50011 2010 Response Spectrum

Calculation of base shear and for response spectrum method in GB 50011-2010.

Details

A three story building is modelled in STAAD.Pro considering the following:

Classification of seismic design = Group 2
Design seismic intensity = 8
Site class = II
Damping = 3.5%

The seismic weight of 420 kN applied as a joint weight of 70 kN to the joints at each floor (joints 2, 3, 4, 6,7, and 8); or 140 kN at each story above ground.

The generated model is subjected to a response spectrum load along global X direction. The base shear and its distribution along height reported by STAAD.Pro is verified against hand calculation.

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	1.114	0.897
2	3.122	0.320
3	4.512	0.222

Table 2. Mode Shapes
Mode	Story Level/Joint Numbers	X-Trans
1	Roof/Joints 4 & 8	1
	2nd Floor/Joints 3 & 7	0.802
	1st Floor/Joints 2 & 6	0.445
2	Roof/Joints 4 & 8	-0.802
	2nd Floor/Joints 3 & 7	0 .445
	1st Floor/Joints 2 & 6	1
3	Roof/Joints 4 & 8	-0.445
	2nd Floor/Joints 3 & 7	1
	1st Floor/Joints 2 & 6	-0.802

Horizontal Acceleration Spectrum Value

The max value of horizontal seismic influence, ɑ_max = 0.16.

The characteristic period, T_g = 0.4s.

The damping ratio, ζ = 0.035.

The power index of the curvilinear decrease section:

= 0.9 + \frac{0.05 - ζ}{0.3 + 6 ζ} = 0.929

The damping adjustment factor is determined as:

η_{2} = 1 + \frac{0.05 - ζ}{0.08 + 1.6 ζ} = 1.11

	Mode 1	Mode 2	Mode 3
Time Period T	0.897 (Tg<T<5Tg)	0.320 (0.1<T < Tg)	0.222 (0.1<T < Tg)
Formula	$ɑ = {(\frac{T_{g}}{T})}^{γ} η_{2} α_{m a x}$	$ɑ = η_{2} α_{m a x}$	$ɑ = η_{2} α_{m a x}$
Seismic Coefficient a	0.0839	0.1776	0.1776

Mode 1

Mode 2

Mode 3

Time Period T

0.897

(Tg<T<5Tg)

0.320

(0.1<T < Tg)

0.222

(0.1<T < Tg)

Formula

ɑ = {(\frac{T_{g}}{T})}^{γ} η_{2} α_{m a x}

ɑ = η_{2} α_{m a x}

ɑ = η_{2} α_{m a x}

Seismic Coefficient a

0.0839

0.1776

Table 3. Mode Participation Factor Calculation
Story Level	Weight Wi (KN)	Mode 1			Mode 2			Mode 3
Story Level	Weight Wi (KN)	ϕ	Wi × ϕ	Wi × ϕ ²	ϕ	Wi × ϕ	Wi × ϕ ²	ϕ	Wi × ϕ	Wi × ϕ ²
Roof	140	1	140	140	-0.802	-112.3	90.05	-0.445	-62.3	27.72
2nd Floor	140	0.802	112.3	90.05	0.445	62.3	27.72	1	140	140
1st Floor	140	0.445	62.3	27.72	1	140	140	-0.802	-112.3	90.05
Summation	420		314.6	257.77		90.02	257.77		-34.58	257.77
Mode Participation Factor Pk ∑Wi × ϕ / ∑ Wi × ϕ ²		1.22			0.349			-0.134
Seismic Coefficient ɑ		0.0839			0.1776			0.1776
Horizontal seismic action	Roof	14.33			-6.959			1.483
	2nd Floor	11.49			3.862			-3.332
	1st Floor	6.377			8.678			2.672
Base shear		32.197			5.581			0.823

Summation by SRSS:

\sqrt{{(32.197)}^{2} + {(5.581)}^{2} + {(0.823)}^{2}} = 32.69

Comparison

Table 4. Comparison of results
Parameter		Hand Calculation	STAAD.Pro	Difference	Comments
Base shear (kN)		32.69	32.67	negligible

Note: The complete quadratic combination method (CQC) in STAAD.Pro gives essentially the same value as the calculation above, as well: 32.69 kN.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\06 Loading\GB 50011\GB 50011 2010 Response Spectrum.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 18-May-15
END JOB INFORMATION
INPUT WIDTH 79
SET SHEAR
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 5.00001 0; 3 0 10 0; 4 0 15 0; 5 5.00001 0 0; 6 5.00001 5.00001 0;
7 5.00001 10 0; 8 5.00001 15 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 5 6; 5 6 7; 6 7 8; 7 2 6; 8 3 7; 9 4 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
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY
1 TO 6 UPTABLE 1 COLLUMN
7 TO 9 UPTABLE 2 BEAM
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 5 FIXED
DEFINE REFERENCE LOADS
LOAD R1 LOADTYPE Mass  TITLE MASS
JOINT LOAD
2 TO 4 6 TO 8 FX 70
END DEFINE REFERENCE LOADS
FLOOR DIAPHRAGM
DIA 1 TYPE RIG HEI 5
DIA 2 TYPE RIG HEI 10
DIA 3 TYPE RIG HEI 15
LOAD 1 LOADTYPE Seismic-H  TITLE EL-X DIR
SPECTRUM CQC GB50011 2010 X 1 ALPHA DAMP 0.035 LOGARITHMIC INTENSITY 8 -
FREQUENT GROUP 2 SCLASS 2
PERFORM ANALYSIS
FINISH

STAAD Output

  GB50011 RESPONSE SPECTRUM LOAD     1 
     GB50011 RESPONSE LOAD CASE      1
            MODAL WEIGHT (MODAL MASS TIMES g) IN KN           GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       3.839138E+02  0.000000E+00  0.000000E+00    2.577637E+02
         2       3.144839E+01  0.000000E+00  0.000000E+00    2.577634E+02
         3       4.638204E+00  0.000000E+00  0.000000E+00    2.577630E+02
     CQC           MODAL COMBINATION METHOD USED.
     DYNAMIC WEIGHT X Y Z   4.200000E+02  0.000000E+00  0.000000E+00 KN  
     MISSING WEIGHT X Y Z   4.013087E-04  0.000000E+00  0.000000E+00 KN  
       MODAL WEIGHT X Y Z   4.200004E+02  0.000000E+00  0.000000E+00 KN  
           MODE                 ACCELERATION-G     DAMPING
           ----                 --------------     -------
              1                      0.08383       0.03500
              2                      0.17765       0.03500
              3                      0.17765       0.03500
      STAAD SPACE                                              -- PAGE NO.    5
 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      0.897       32.18        0.00        0.00        0.00        0.00     -361.56
      2      0.320        5.59        0.00        0.00        0.00        0.00       22.40
      3      0.222        0.82        0.00        0.00        0.00        0.00       -2.29
 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   91.41  0.00  0.00   91.408    0.000    0.000     32.18     0.00     0.00
   2    7.49  0.00  0.00   98.896    0.000    0.000      5.59     0.00     0.00
   3    1.10  0.00  0.00  100.000    0.000    0.000      0.82     0.00     0.00
                                                    ---------------------------
                                  TOTAL CQC   SHEAR     32.68     0.00     0.00