V. Floor Spectrum Verification 1
Calculate the response spectrum for a ramp acceleration time history.
Reference
- Biggs, John M., Introduction to Structural Dynamics, McGraw-Hill, 1964, pp. 256-258.
- Kelly, R. D. and Richman, G., Principle and Techniques of Shock Data Analysis, The Shock and Vibration Information Center, Washington D.C., 1969.
- Smallwood, David O, An Improved Recursive Formula for Calculating Shock Response Spectra, Shock and Vibration Bulletin, No. 51, May 1981.
- Tom Irvine (2023). Shock Response Spectrum, MATLAB Central File Exchange. Retrieved February 7, 2023.
- Wenjuan Sun, Qiaoping Su, Hongli Yuan and Liqin Liu,Calculation and Characteristic Analysis on Different Types of Shock Response Spectrum, Journal of Physics: Conference Series 1676 (2020).
Details
The purpose of this example is to compare the theoretical acceleration spectral values for the specified motion with those computed by STAAD.Pro.
Figure 1: Ideal Acceleration Time History
where:
- td = 0.1 sec
- ӱso = 1.0 g
The acceleration time history is approximated by the curve shown below using a time interval of 0.001 seconds. The response of the system is computed for a time up to 0.250 seconds to account for the transient response to the impulse.
Figure 2: Input Acceleration Time History
Figure 3: Vertical Cantilever Model: divide into three segments at 10’, 5’, and 5’
- E = 4.536x108 psf
- L = 20 ft
- I = 1/12 ft4
- A = 1 ft2
- Material Density = 149.99 lbf/ft3
- Damping Ratio = 5%
- g = 32.172 ft/sec2
The above model was subjected to the ground motion described in Fig. 2. above. Note that the absolute time history acceleration response at the ground node will be exactly the same as the input ground motion. Thus, the floor spectrum values calculated by STAAD.Pro at the ground node are the acceleration spectral values corresponding to the input ground motion.
Validation
Methodology
As STAAD.Pro requires non-zero damping to be specified while calculating the floor spectrum, it is not possible to directly compare the STAAD.Pro generated floor spectrum with the theoretical solution with zero damping parameter as described in the next section. In lieu of a direct comparison, first, alternative methods to compute the acceleration spectrum per Kelly-Richman (2) and Smallwood (3) are developed by referring to the MATLAB® code developed by Irvine (4). The algorithm prosed by Smallwood is widely accepted in the industry due to its computational accuracy (5). The result from the Smallwood (3) method is compared against the theoretical solution (1) to establish the validity of the alternative method.
Following that, the floor spectrum or the spectral acceleration for the acceleration time-history per Fig. 2 is computed using STAAD.Pro for a 5% damping ratio. The same problem is also solved using the Smallwood algorithm. The results from STAAD.Pro and this alternative method are compared to establish the validity of the STAAD.Pro results.Theoretical Solution
A theoretical solution based on the Reference 1 is developed for the system without damping.
The equation of motion for a damped single degree of freedom system subjected to a base acceleration of ӱso fa(t) is:
mü + ců + ku = -m ӱso fa(t)
Where u = u(t) = relative displacement of the mass with respect to the support. This equation can be expressed as:
ü + 2 βů + ω2u = - ӱso fa(t)
The response of the system is given by:
意味| = | ||
| = | ||
| = | ||
| = |
For this example:
- ӱso = 1.0 g
- β = 0.0
- for t ≤ 2td
- for t > 2td
Therefore, the equation for the dynamic load factor (DLF) becomes:
When integrated, this equation becomes (Eqn. 6.14(1)):
After time 2td, the equation for the dynamic load factor is from the reference (Eqn. 6 .15(1)):
which is based upon consideration of the DLF and its derivative at t = 2td as an initial displacement and velocity for the motion following the time 2td.
As per the Eqn. 6.13(1), the maximum acceleration response is |ӱso (DLF)max.|
The maximum value of the dynamic load factor according to these two equations was evaluated using a time interval of 0.001 seconds. The theoretical values (1) of the peak or spectral acceleration at a few discrete frequencies are listed in Table 1 below. The range of the peak values are plotted in Fig. 4.
| Frequency (Hz·) | Theoretical |
|---|---|
| 5.000 | 2.000 |
| 6.000 | 1.667 |
| 6.098 | 1.640 |
| 7.000 | 1.429 |
| 8.000 | 1.453 |
| 9.000 | 1.834 |
| 10.000 | 2.000 |
| 11.000 | 1.818 |
| 12.000 | 1.666 |
| 13.000 | 1.644 |
| 14.000 | 1.762 |
| 15.000 | 2.000 |
| 20.000 | 2.001 |
| 35.000 | 2.003 |
| 50.000 | 2.005 |
Validation of Alternative Method
As the alternate methods, a MATLAB / Octave code was developed based on the reference (4) to compute acceleration spectra using Kelly-Richman (2) and Smallwood (3) algorithms to calculate the acceleration spectra for given acceleration time history. The acceleration spectra are calculated by the alternate method following the Smallwood algorithm and compared with the theoretical solution as presented in Fig. 4.
The plots in Fig 4 suggest that the alternative method agrees with the theoretical values, validating the alternative methods.
Results
Using the GENERATE FLOOR SPECTRUM command set in STAAD.Pro, the floor spectrum is generated for a damping ratio of 5%. The frequency vs. spectral acceleration pairs is extracted from the SPRINT output as listed below in the STAAD Output section below. The floor spectrum is also calculated for the same damping ratio using the Smallwood algorithm using the validated alternative method. The spectral curve for the STAAD.Pro floor spectrum results is plotted and compared with the spectrum produced by the Smallwood algorithm. As demonstrated in Fig. 5 below, STAAD.Pro generated floor spectrum is in excellent agreement with the alternative method, thus validating the floor spectrum generation using STAAD.Pro.
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\08 Dynamic Analysis\Floor Spectrum Verification 1.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 03-Feb-23
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET POUND
JOINT COORDINATES
1 0 0 0; 2 0 20 0; 3 0 10 0; 4 0 15 0;
MEMBER INCIDENCES
1 1 3; 2 3 4; 3 4 2;
START GROUP DEFINITION
JOINT
_GROUND 1
END GROUP DEFINITION
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 4.536e+08
POISSON 0.17
DENSITY 149.99
ALPHA 5.5e-06
DAMP 0.05
G 1.93846e+08
TYPE CONCRETE
STRENGTH FCU 576000
END DEFINE MATERIAL
MEMBER PROPERTY
1 TO 3 PRIS YD 1
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 FIXED
DEFINE REFERENCE LOADS
LOAD R1 LOADTYPE Mass TITLE MASS
JOINT LOAD
2 3 FX 3.0912 FY 3.0912 FZ 3.0912
4 FX 1.5456 FY 1.5456 FZ 1.5456
END DEFINE REFERENCE LOADS
DEFINE TIME HISTORY DT 0.001
TYPE 1 ACCELERATION
0 0 0.001 1 0.199 -1 0.2 0 0.25 0
ARRIVAL TIME
0
DAMPING 0.05
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
GROUND MOTION ABS X 1 1 32.172
PERFORM ANALYSIS
GENERATE FLOOR SPECTRUM
BEGIN FLOOR DIRECTION GX AT GROUND LEVEL
_GROUND
OPTIONS FLOW 0.500 FHIGH 60.000 FDELTA 0.100 DAMP 0.050 THPRINT 2 SPRINT
END FLOOR SPECTRUM
FINISH
STAAD.Pro Output
FLOOR SPECTRUMS EVALUATION OF ABSOLUTE ACCELERATION HISTORY AT FLOOR NUMBER 1 X DIRECTION VALUES IN G UNITS MAX VALUE 9.999364E-01 MIN VALUE -9.999279E-01 MEAN VALUE 1.716036E-10 RMS VALUE 5.176672E-01 EXPECTED FREQUENCY 1.701907E+01 HZ OUTPUT ABSOLUTE ACCELERATION SPECTRUM FLOOR NUMBER 1 X DIRECTION DAMPING= 0.050 TITLE: AT GROUND LEVEL POINT FREQUENCY DISPLACEMENT VELOCITY ACCEL G PK TIME (ACCEL) 1 5.000000E-01 2.350797E+00 1.848440E+01 6.409103E-02 1.640000E-01 2 6.000000E-01 2.284843E+00 1.823932E+01 8.822565E-02 1.650000E-01 3 7.000000E-01 2.214131E+00 1.797431E+01 1.151403E-01 1.650000E-01 4 8.000000E-01 2.144662E+00 1.769483E+01 1.443165E-01 1.640000E-01 5 9.000000E-01 2.130347E+00 1.740297E+01 1.773300E-01 2.480000E-01 6 1.000000E+00 2.114422E+00 1.710208E+01 2.172897E-01 2.480000E-01 7 1.100000E+00 2.096922E+00 1.679510E+01 2.607452E-01 2.480000E-01 8 1.200000E+00 2.077874E+00 1.699101E+01 3.074888E-01 2.480000E-01 9 1.300000E+00 2.057311E+00 1.822487E+01 3.573007E-01 2.480000E-01 10 1.400000E+00 2.035274E+00 1.941637E+01 4.099449E-01 2.480000E-01 11 1.500000E+00 2.011784E+00 2.056326E+01 4.651660E-01 2.480000E-01 12 1.600000E+00 1.986888E+00 2.166247E+01 5.227106E-01 2.480000E-01 13 1.700000E+00 1.960622E+00 2.271232E+01 5.822906E-01 2.480000E-01 14 1.800000E+00 1.933019E+00 2.371009E+01 6.436194E-01 2.480000E-01 15 1.900000E+00 1.904165E+00 2.465338E+01 7.064200E-01 2.480000E-01 16 2.000000E+00 1.874070E+00 2.554060E+01 7.703610E-01 2.480000E-01 17 2.100000E+00 1.842810E+00 2.637056E+01 8.351448E-01 2.480000E-01 18 2.200000E+00 1.810402E+00 2.714028E+01 9.004525E-01 2.480000E-01 19 2.300000E+00 1.776918E+00 2.784952E+01 9.659824E-01 2.480000E-01 20 2.400000E+00 1.742385E+00 2.844258E+01 1.031393E+00 2.480000E-01 21 2.500000E+00 1.706908E+00 2.886397E+01 1.096345E+00 2.480000E-01 22 2.600000E+00 1.670557E+00 2.910931E+01 1.160533E+00 2.480000E-01 23 2.700000E+00 1.633294E+00 2.917632E+01 1.223588E+00 2.480000E-01 24 2.800000E+00 1.595285E+00 2.906489E+01 1.285273E+00 2.480000E-01 25 2.900000E+00 1.556508E+00 2.877695E+01 1.345200E+00 2.480000E-01 26 3.000000E+00 1.517031E+00 2.835353E+01 1.403091E+00 2.480000E-01 27 3.100000E+00 1.476939E+00 2.776677E+01 1.458614E+00 2.480000E-01 28 3.200000E+00 1.436366E+00 2.702173E+01 1.511520E+00 2.480000E-01 29 3.300000E+00 1.395266E+00 2.680904E+01 1.561431E+00 2.430000E-01 30 3.400000E+00 1.353751E+00 2.679938E+01 1.608178E+00 2.390000E-01 31 3.500000E+00 1.311854E+00 2.673381E+01 1.651373E+00 2.350000E-01 32 3.600000E+00 1.269629E+00 2.661257E+01 1.690974E+00 2.320000E-01 33 3.700000E+00 1.227200E+00 2.643817E+01 1.726433E+00 2.280000E-01 34 3.800000E+00 1.184546E+00 2.621007E+01 1.757854E+00 2.250000E-01 35 3.900000E+00 1.141855E+00 2.592859E+01 1.784815E+00 2.220000E-01 36 4.000000E+00 1.099075E+00 2.559824E+01 1.807149E+00 2.190000E-01 37 4.100000E+00 1.056263E+00 2.521568E+01 1.824684E+00 2.170000E-01 STAAD SPACE -- PAGE NO. 7 38 4.200000E+00 1.013614E+00 2.478644E+01 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7.893734E+00 1.382789E+00 4.900000E-02 83 8.700000E+00 1.787450E-01 8.165838E+00 1.387764E+00 4.800000E-02 84 8.800000E+00 1.752672E-01 8.405481E+00 1.392369E+00 4.800000E-02 85 8.900000E+00 1.719586E-01 8.612009E+00 1.397103E+00 4.700000E-02 86 9.000000E+00 1.686662E-01 8.785094E+00 1.401612E+00 4.700000E-02 87 9.100000E+00 1.684909E-01 8.924697E+00 1.433812E+00 2.050000E-01 88 9.200000E+00 1.686933E-01 9.034207E+00 1.467064E+00 2.040000E-01 89 9.300000E+00 1.682815E-01 9.112021E+00 1.495218E+00 2.030000E-01 90 9.400000E+00 1.672761E-01 9.157156E+00 1.518977E+00 2.030000E-01 91 9.500000E+00 1.657210E-01 9.170222E+00 1.537526E+00 2.020000E-01 92 9.600000E+00 1.637085E-01 9.151957E+00 1.550652E+00 2.010000E-01 93 9.700000E+00 1.611790E-01 9.103197E+00 1.558221E+00 2.000000E-01 STAAD SPACE -- PAGE NO. 8 94 9.800000E+00 1.581539E-01 9.024856E+00 1.560416E+00 2.000000E-01 95 9.900000E+00 1.546542E-01 8.917899E+00 1.558154E+00 1.990000E-01 96 1.000000E+01 1.508253E-01 8.783329E+00 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4.456811E+00 1.529537E+00 3.400000E-02 126 1.300000E+01 8.839414E-02 4.419036E+00 1.531480E+00 3.400000E-02 127 1.310000E+01 8.715568E-02 4.379609E+00 1.533979E+00 3.300000E-02 128 1.320000E+01 8.598651E-02 4.340624E+00 1.536598E+00 3.300000E-02 129 1.330000E+01 8.483916E-02 4.304462E+00 1.538863E+00 3.300000E-02 130 1.340000E+01 8.369595E-02 4.264780E+00 1.540773E+00 3.300000E-02 131 1.350000E+01 8.255702E-02 4.230838E+00 1.542770E+00 3.200000E-02 132 1.360000E+01 8.146535E-02 4.362512E+00 1.545396E+00 3.200000E-02 133 1.370000E+01 8.041218E-02 4.531747E+00 1.547689E+00 3.200000E-02 134 1.380000E+01 7.936270E-02 4.684374E+00 1.549646E+00 3.200000E-02 135 1.390000E+01 7.831702E-02 4.825470E+00 1.551267E+00 3.200000E-02 136 1.400000E+01 7.728008E-02 4.949535E+00 1.553545E+00 3.100000E-02 137 1.410000E+01 7.631516E-02 5.060318E+00 1.555939E+00 3.100000E-02 138 1.420000E+01 7.535348E-02 5.151950E+00 1.558018E+00 3.100000E-02 139 1.430000E+01 7.439515E-02 5.231017E+00 1.559781E+00 3.100000E-02 140 1.440000E+01 7.344027E-02 5.287983E+00 1.561227E+00 3.100000E-02 141 1.450000E+01 7.250961E-02 5.334596E+00 1.563479E+00 3.000000E-02 142 1.460000E+01 7.163017E-02 5.356190E+00 1.565746E+00 3.000000E-02 143 1.470000E+01 7.075364E-02 5.370164E+00 1.567716E+00 3.000000E-02 144 1.480000E+01 6.988012E-02 5.357053E+00 1.569390E+00 3.000000E-02 145 1.490000E+01 6.900968E-02 5.338645E+00 1.570766E+00 3.000000E-02 146 1.500000E+01 6.815722E-02 5.292682E+00 1.572626E+00 2.900000E-02 147 1.510000E+01 6.735732E-02 5.242482E+00 1.574863E+00 2.900000E-02 148 1.520000E+01 6.655998E-02 5.166496E+00 1.576821E+00 2.900000E-02 149 1.530000E+01 6.576529E-02 5.085330E+00 1.578501E+00 2.900000E-02 STAAD SPACE -- PAGE NO. 9 150 1.540000E+01 6.497333E-02 4.982917E+00 1.579902E+00 2.900000E-02 151 1.550000E+01 6.418419E-02 4.871768E+00 1.581024E+00 2.900000E-02 152 1.560000E+01 6.344741E-02 4.747072E+00 1.583239E+00 2.800000E-02 153 1.570000E+01 6.272367E-02 4.607014E+00 1.585273E+00 2.800000E-02 154 1.580000E+01 6.200225E-02 4.464526E+00 1.587047E+00 2.800000E-02 155 1.590000E+01 6.128320E-02 4.307627E+00 1.588561E+00 2.800000E-02 156 1.600000E+01 6.056660E-02 4.141053E+00 1.589813E+00 2.800000E-02 157 1.610000E+01 5.985494E-02 3.974769E+00 1.590804E+00 2.800000E-02 158 1.620000E+01 5.919959E-02 3.800802E+00 1.592918E+00 2.700000E-02 159 1.630000E+01 5.854622E-02 3.622177E+00 1.594864E+00 2.700000E-02 160 1.640000E+01 5.789490E-02 3.441920E+00 1.596568E+00 2.700000E-02 161 1.650000E+01 5.724568E-02 3.365042E+00 1.598027E+00 2.700000E-02 162 1.660000E+01 5.659860E-02 3.342886E+00 1.599243E+00 2.700000E-02 163 1.670000E+01 5.595374E-02 3.322160E+00 1.600216E+00 2.700000E-02 164 1.680000E+01 5.535588E-02 3.299595E+00 1.601691E+00 2.600000E-02 165 1.690000E+01 5.476747E-02 3.275228E+00 1.603652E+00 2.600000E-02 166 1.700000E+01 5.418083E-02 3.254692E+00 1.605387E+00 2.600000E-02 167 1.710000E+01 5.359601E-02 3.235043E+00 1.606895E+00 2.600000E-02 168 1.720000E+01 5.301306E-02 3.213676E+00 1.608177E+00 2.600000E-02 169 1.730000E+01 5.243202E-02 3.190625E+00 1.609232E+00 2.600000E-02 170 1.740000E+01 5.186308E-02 3.170465E+00 1.610060E+00 2.600000E-02 171 1.750000E+01 5.133458E-02 3.152170E+00 1.611513E+00 2.500000E-02 172 1.760000E+01 5.080760E-02 3.132267E+00 1.613371E+00 2.500000E-02 173 1.770000E+01 5.028217E-02 3.110787E+00 1.615020E+00 2.500000E-02 174 1.780000E+01 4.975834E-02 3.089715E+00 1.616458E+00 2.500000E-02 175 1.790000E+01 4.923614E-02 3.073012E+00 1.617687E+00 2.500000E-02 176 1.800000E+01 4.871561E-02 3.054801E+00 1.618704E+00 2.500000E-02 177 1.810000E+01 4.819913E-02 3.035109E+00 1.619511E+00 2.500000E-02 178 1.820000E+01 4.772716E-02 3.013966E+00 1.620248E+00 2.400000E-02 179 1.830000E+01 4.725650E-02 2.996890E+00 1.622117E+00 2.400000E-02 180 1.840000E+01 4.678717E-02 2.980565E+00 1.623791E+00 2.400000E-02 181 1.850000E+01 4.631921E-02 2.962848E+00 1.625272E+00 2.400000E-02 182 1.860000E+01 4.585265E-02 3.007060E+00 1.626557E+00 2.400000E-02 183 1.870000E+01 4.538753E-02 3.104534E+00 1.627648E+00 2.400000E-02 184 1.880000E+01 4.492387E-02 3.194961E+00 1.628544E+00 2.400000E-02 185 1.890000E+01 4.447532E-02 3.282295E+00 1.629246E+00 2.400000E-02 186 1.900000E+01 4.405626E-02 3.365036E+00 1.629752E+00 2.400000E-02 187 1.910000E+01 4.363832E-02 3.437045E+00 1.631538E+00 2.300000E-02 188 1.920000E+01 4.322153E-02 3.496894E+00 1.633159E+00 2.300000E-02 189 1.930000E+01 4.280592E-02 3.553328E+00 1.634600E+00 2.300000E-02 190 1.940000E+01 4.239152E-02 3.601199E+00 1.635863E+00 2.300000E-02 191 1.950000E+01 4.197835E-02 3.634928E+00 1.636946E+00 2.300000E-02 192 1.960000E+01 4.156644E-02 3.657788E+00 1.637849E+00 2.300000E-02 193 1.970000E+01 4.115582E-02 3.678445E+00 1.638572E+00 2.300000E-02 194 1.980000E+01 4.078330E-02 3.684211E+00 1.639116E+00 2.300000E-02 195 1.990000E+01 4.041337E-02 3.675044E+00 1.639753E+00 2.200000E-02 196 2.000000E+01 4.004442E-02 3.665557E+00 1.641435E+00 2.200000E-02 197 2.010000E+01 3.967647E-02 3.643973E+00 1.642954E+00 2.200000E-02 198 2.020000E+01 3.930953E-02 3.608374E+00 1.644307E+00 2.200000E-02 199 2.030000E+01 3.894363E-02 3.564406E+00 1.645496E+00 2.200000E-02 200 2.040000E+01 3.857879E-02 3.518175E+00 1.646519E+00 2.200000E-02 201 2.050000E+01 3.821503E-02 3.459757E+00 1.647378E+00 2.200000E-02 202 2.060000E+01 3.785238E-02 3.390245E+00 1.648072E+00 2.200000E-02 203 2.070000E+01 3.752133E-02 3.314480E+00 1.648600E+00 2.200000E-02 204 2.080000E+01 3.719592E-02 3.238240E+00 1.648964E+00 2.200000E-02 205 2.090000E+01 3.687133E-02 3.153697E+00 1.650028E+00 2.100000E-02 STAAD SPACE -- PAGE NO. 10 206 2.100000E+01 3.654758E-02 3.062332E+00 1.651572E+00 2.100000E-02 207 2.110000E+01 3.622469E-02 2.965711E+00 1.652965E+00 2.100000E-02 208 2.120000E+01 3.590267E-02 2.865448E+00 1.654208E+00 2.100000E-02 209 2.130000E+01 3.558154E-02 2.764471E+00 1.655300E+00 2.100000E-02 210 2.140000E+01 3.526131E-02 2.665436E+00 1.656241E+00 2.100000E-02 211 2.150000E+01 3.494201E-02 2.566009E+00 1.657031E+00 2.100000E-02 212 2.160000E+01 3.462365E-02 2.494065E+00 1.657670E+00 2.100000E-02 213 2.170000E+01 3.433677E-02 2.483238E+00 1.658157E+00 2.100000E-02 214 2.180000E+01 3.405232E-02 2.471548E+00 1.658494E+00 2.100000E-02 215 2.190000E+01 3.376856E-02 2.459007E+00 1.658769E+00 2.000000E-02 216 2.200000E+01 3.348551E-02 2.445628E+00 1.660314E+00 2.000000E-02 217 2.210000E+01 3.320317E-02 2.431426E+00 1.661721E+00 2.000000E-02 218 2.220000E+01 3.292157E-02 2.418347E+00 1.662992E+00 2.000000E-02 219 2.230000E+01 3.264070E-02 2.408849E+00 1.664124E+00 2.000000E-02 220 2.240000E+01 3.236060E-02 2.398552E+00 1.665120E+00 2.000000E-02 221 2.250000E+01 3.208127E-02 2.387467E+00 1.665978E+00 2.000000E-02 222 2.260000E+01 3.180272E-02 2.375607E+00 1.666698E+00 2.000000E-02 223 2.270000E+01 3.152748E-02 2.362982E+00 1.667280E+00 2.000000E-02 224 2.280000E+01 3.127990E-02 2.349606E+00 1.667725E+00 2.000000E-02 225 2.290000E+01 3.103288E-02 2.337135E+00 1.668033E+00 2.000000E-02 226 2.300000E+01 3.078644E-02 2.328412E+00 1.668203E+00 2.000000E-02 227 2.310000E+01 3.054059E-02 2.318958E+00 1.668979E+00 1.900000E-02 228 2.320000E+01 3.029533E-02 2.308785E+00 1.670396E+00 1.900000E-02 229 2.330000E+01 3.005068E-02 2.297901E+00 1.671689E+00 1.900000E-02 230 2.340000E+01 2.980664E-02 2.286319E+00 1.672857E+00 1.900000E-02 231 2.350000E+01 2.956323E-02 2.274047E+00 1.673901E+00 1.900000E-02 232 2.360000E+01 2.932047E-02 2.315262E+00 1.674819E+00 1.900000E-02 233 2.370000E+01 2.907835E-02 2.372272E+00 1.675614E+00 1.900000E-02 234 2.380000E+01 2.883689E-02 2.426706E+00 1.676283E+00 1.900000E-02 235 2.390000E+01 2.859986E-02 2.477414E+00 1.676827E+00 1.900000E-02 236 2.400000E+01 2.838636E-02 2.529942E+00 1.677247E+00 1.900000E-02 237 2.410000E+01 2.817332E-02 2.578645E+00 1.677542E+00 1.900000E-02 238 2.420000E+01 2.796075E-02 2.621805E+00 1.677713E+00 1.900000E-02 239 2.430000E+01 2.774867E-02 2.658583E+00 1.677758E+00 1.900000E-02 240 2.440000E+01 2.753707E-02 2.688269E+00 1.678758E+00 1.800000E-02 241 2.450000E+01 2.732597E-02 2.710288E+00 1.680086E+00 1.800000E-02 242 2.460000E+01 2.711538E-02 2.730300E+00 1.681302E+00 1.800000E-02 243 2.470000E+01 2.690531E-02 2.747874E+00 1.682405E+00 1.800000E-02 244 2.480000E+01 2.669575E-02 2.757220E+00 1.683396E+00 1.800000E-02 245 2.490000E+01 2.648673E-02 2.758180E+00 1.684274E+00 1.800000E-02 246 2.500000E+01 2.627825E-02 2.750745E+00 1.685040E+00 1.800000E-02 247 2.510000E+01 2.607031E-02 2.735096E+00 1.685693E+00 1.800000E-02 248 2.520000E+01 2.586293E-02 2.723400E+00 1.686234E+00 1.800000E-02 249 2.530000E+01 2.567873E-02 2.703756E+00 1.686662E+00 1.800000E-02 250 2.540000E+01 2.549633E-02 2.676529E+00 1.686978E+00 1.800000E-02 251 2.550000E+01 2.531431E-02 2.642207E+00 1.687181E+00 1.800000E-02 252 2.560000E+01 2.513268E-02 2.601385E+00 1.687272E+00 1.800000E-02 253 2.570000E+01 2.495145E-02 2.554754E+00 1.687250E+00 1.800000E-02 254 2.580000E+01 2.477062E-02 2.509748E+00 1.687894E+00 1.700000E-02 255 2.590000E+01 2.459020E-02 2.461723E+00 1.689182E+00 1.700000E-02 256 2.600000E+01 2.441019E-02 2.409361E+00 1.690371E+00 1.700000E-02 257 2.610000E+01 2.423061E-02 2.353530E+00 1.691458E+00 1.700000E-02 258 2.620000E+01 2.405145E-02 2.295138E+00 1.692445E+00 1.700000E-02 259 2.630000E+01 2.387273E-02 2.235118E+00 1.693330E+00 1.700000E-02 260 2.640000E+01 2.369445E-02 2.174409E+00 1.694115E+00 1.700000E-02 261 2.650000E+01 2.351661E-02 2.113944E+00 1.694799E+00 1.700000E-02 STAAD SPACE -- PAGE NO. 11 262 2.660000E+01 2.333923E-02 2.054630E+00 1.695382E+00 1.700000E-02 263 2.670000E+01 2.316830E-02 2.017403E+00 1.695865E+00 1.700000E-02 264 2.680000E+01 2.301335E-02 2.010678E+00 1.696246E+00 1.700000E-02 265 2.690000E+01 2.285870E-02 2.003944E+00 1.696526E+00 1.700000E-02 266 2.700000E+01 2.270437E-02 1.997201E+00 1.696706E+00 1.700000E-02 267 2.710000E+01 2.255034E-02 1.990449E+00 1.696785E+00 1.700000E-02 268 2.720000E+01 2.239664E-02 1.983687E+00 1.696763E+00 1.700000E-02 269 2.730000E+01 2.224326E-02 1.976917E+00 1.696640E+00 1.700000E-02 270 2.740000E+01 2.209020E-02 1.970137E+00 1.697401E+00 1.600000E-02 271 2.750000E+01 2.193749E-02 1.963349E+00 1.698619E+00 1.600000E-02 272 2.760000E+01 2.178511E-02 1.956553E+00 1.699748E+00 1.600000E-02 273 2.770000E+01 2.163307E-02 1.949748E+00 1.700788E+00 1.600000E-02 274 2.780000E+01 2.148139E-02 1.942934E+00 1.701737E+00 1.600000E-02 275 2.790000E+01 2.133005E-02 1.936112E+00 1.702597E+00 1.600000E-02 276 2.800000E+01 2.117907E-02 1.929282E+00 1.703367E+00 1.600000E-02 277 2.810000E+01 2.102846E-02 1.922444E+00 1.704047E+00 1.600000E-02 278 2.820000E+01 2.087821E-02 1.915598E+00 1.704637E+00 1.600000E-02 279 2.830000E+01 2.072833E-02 1.908745E+00 1.705137E+00 1.600000E-02 280 2.840000E+01 2.059595E-02 1.901883E+00 1.705547E+00 1.600000E-02 281 2.850000E+01 2.046592E-02 1.895014E+00 1.705868E+00 1.600000E-02 282 2.860000E+01 2.033613E-02 1.900092E+00 1.706098E+00 1.600000E-02 283 2.870000E+01 2.020659E-02 1.934873E+00 1.706239E+00 1.600000E-02 284 2.880000E+01 2.007730E-02 1.969295E+00 1.706290E+00 1.600000E-02 285 2.890000E+01 1.994826E-02 2.002640E+00 1.706251E+00 1.600000E-02 286 2.900000E+01 1.981948E-02 2.034220E+00 1.706123E+00 1.600000E-02 287 2.910000E+01 1.969096E-02 2.063391E+00 1.705905E+00 1.600000E-02 288 2.920000E+01 1.956271E-02 2.089562E+00 1.706820E+00 1.500000E-02 289 2.930000E+01 1.943473E-02 2.112204E+00 1.707973E+00 1.500000E-02 290 2.940000E+01 1.930702E-02 2.130859E+00 1.709046E+00 1.500000E-02 291 2.950000E+01 1.917959E-02 2.145145E+00 1.710041E+00 1.500000E-02 292 2.960000E+01 1.905243E-02 2.160689E+00 1.710956E+00 1.500000E-02 293 2.970000E+01 1.892556E-02 2.174158E+00 1.711791E+00 1.500000E-02 294 2.980000E+01 1.879898E-02 2.182887E+00 1.712547E+00 1.500000E-02 295 2.990000E+01 1.867268E-02 2.186714E+00 1.713224E+00 1.500000E-02 296 3.000000E+01 1.854668E-02 2.185560E+00 1.713821E+00 1.500000E-02 297 3.010000E+01 1.842098E-02 2.179430E+00 1.714339E+00 1.500000E-02 298 3.020000E+01 1.830338E-02 2.168415E+00 1.714777E+00 1.500000E-02 299 3.030000E+01 1.819523E-02 2.152682E+00 1.715136E+00 1.500000E-02 300 3.040000E+01 1.808728E-02 2.132508E+00 1.715415E+00 1.500000E-02 301 3.050000E+01 1.797951E-02 2.116323E+00 1.715615E+00 1.500000E-02 302 3.060000E+01 1.787193E-02 2.095884E+00 1.715735E+00 1.500000E-02 303 3.070000E+01 1.776456E-02 2.071534E+00 1.715776E+00 1.500000E-02 304 3.080000E+01 1.765738E-02 2.043673E+00 1.715738E+00 1.500000E-02 305 3.090000E+01 1.755041E-02 2.012753E+00 1.715621E+00 1.500000E-02 306 3.100000E+01 1.744364E-02 1.979264E+00 1.715424E+00 1.500000E-02 307 3.110000E+01 1.733708E-02 1.943733E+00 1.715148E+00 1.500000E-02 308 3.120000E+01 1.723072E-02 1.906708E+00 1.715670E+00 1.400000E-02 309 3.130000E+01 1.712459E-02 1.868754E+00 1.716790E+00 1.400000E-02 310 3.140000E+01 1.701866E-02 1.830436E+00 1.717840E+00 1.400000E-02 311 3.150000E+01 1.691296E-02 1.792318E+00 1.718820E+00 1.400000E-02 312 3.160000E+01 1.680747E-02 1.754947E+00 1.719731E+00 1.400000E-02 313 3.170000E+01 1.670221E-02 1.718848E+00 1.720573E+00 1.400000E-02 314 3.180000E+01 1.659717E-02 1.709620E+00 1.721344E+00 1.400000E-02 315 3.190000E+01 1.649237E-02 1.704276E+00 1.722046E+00 1.400000E-02 316 3.200000E+01 1.638779E-02 1.698927E+00 1.722678E+00 1.400000E-02 317 3.210000E+01 1.628344E-02 1.693572E+00 1.723241E+00 1.400000E-02 STAAD SPACE -- PAGE NO. 12 318 3.220000E+01 1.617933E-02 1.688211E+00 1.723734E+00 1.400000E-02 319 3.230000E+01 1.608399E-02 1.682846E+00 1.724157E+00 1.400000E-02 320 3.240000E+01 1.599524E-02 1.677475E+00 1.724510E+00 1.400000E-02 321 3.250000E+01 1.590663E-02 1.672099E+00 1.724794E+00 1.400000E-02 322 3.260000E+01 1.581817E-02 1.666718E+00 1.725008E+00 1.400000E-02 323 3.270000E+01 1.572986E-02 1.661333E+00 1.725153E+00 1.400000E-02 324 3.280000E+01 1.564170E-02 1.655942E+00 1.725228E+00 1.400000E-02 325 3.290000E+01 1.555369E-02 1.650547E+00 1.725233E+00 1.400000E-02 326 3.300000E+01 1.546584E-02 1.645146E+00 1.725169E+00 1.400000E-02 327 3.310000E+01 1.537814E-02 1.639741E+00 1.725035E+00 1.400000E-02 328 3.320000E+01 1.529061E-02 1.634332E+00 1.724832E+00 1.400000E-02 329 3.330000E+01 1.520323E-02 1.628918E+00 1.724560E+00 1.400000E-02 330 3.340000E+01 1.511601E-02 1.623499E+00 1.724218E+00 1.400000E-02 331 3.350000E+01 1.502896E-02 1.618076E+00 1.724513E+00 1.300000E-02 332 3.360000E+01 1.494208E-02 1.614139E+00 1.725591E+00 1.300000E-02 333 3.370000E+01 1.485536E-02 1.634461E+00 1.726609E+00 1.300000E-02 334 3.380000E+01 1.476881E-02 1.655727E+00 1.727566E+00 1.300000E-02 335 3.390000E+01 1.468243E-02 1.676950E+00 1.728463E+00 1.300000E-02 336 3.400000E+01 1.459623E-02 1.697690E+00 1.729299E+00 1.300000E-02 337 3.410000E+01 1.451020E-02 1.717524E+00 1.730075E+00 1.300000E-02 338 3.420000E+01 1.442435E-02 1.736054E+00 1.730790E+00 1.300000E-02 339 3.430000E+01 1.433868E-02 1.752909E+00 1.731445E+00 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1.741438E+00 1.200000E-02 369 3.730000E+01 1.217872E-02 1.466270E+00 1.741932E+00 1.200000E-02 370 3.740000E+01 1.211198E-02 1.462222E+00 1.742375E+00 1.200000E-02 371 3.750000E+01 1.205476E-02 1.458170E+00 1.742766E+00 1.200000E-02 372 3.760000E+01 1.199762E-02 1.454115E+00 1.743105E+00 1.200000E-02 373 3.770000E+01 1.194056E-02 1.450057E+00 1.743392E+00 1.200000E-02 STAAD SPACE -- PAGE NO. 13 374 3.780000E+01 1.188358E-02 1.445996E+00 1.743627E+00 1.200000E-02 375 3.790000E+01 1.182668E-02 1.441931E+00 1.743810E+00 1.200000E-02 376 3.800000E+01 1.176987E-02 1.437864E+00 1.743941E+00 1.200000E-02 377 3.810000E+01 1.171314E-02 1.433793E+00 1.744020E+00 1.200000E-02 378 3.820000E+01 1.165649E-02 1.429719E+00 1.744047E+00 1.200000E-02 379 3.830000E+01 1.159994E-02 1.425643E+00 1.744023E+00 1.200000E-02 380 3.840000E+01 1.154346E-02 1.421563E+00 1.743946E+00 1.200000E-02 381 3.850000E+01 1.148708E-02 1.417481E+00 1.743818E+00 1.200000E-02 382 3.860000E+01 1.143079E-02 1.416653E+00 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