Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 0
EXAMPLE 1-014 FRAME - EIGENVALUE PROBLEM EXAMPLE DESCRIPTION The SAP2000 eigenvalue computations are verified using vibrations of a cantilever beam. This example uses several models of an eight foot long cantilever concrete beam with I22 ď‚š I33. Each of the models has a different discretization. The first five bending Eigen modes for each model are compared with the independent solution provided in Clough and Penzien 1975. Important Note: Only bending modes are calculated and compared. Shear deformations are ignored by setting the frame property modification factor for shear area to zero. Axial and torsional modes are excluded by excluding the Ux and Rx degrees of freedom from the analysis. GEOMETRY AND PROPERTIES Discretization Model A: 1 element 96 inches long Model B: 2 elements each 48 inches long Model C: 4 elements each 24 inches long Model D: 6 elements each 16 inches long Model E: 8 elements each 12 inch long Model F: 10 elements each 9.6 inches long Model G: 96 elements each 1 inch long
Z Y
96" X Material Properties E = 3,600 k/in2 Mass per unit volume = 2.3E-07 k-sec2/in4
Section Properties b = 12 in d = 18 in A = 216 in2 I about global Y = 5,832 in4 I about global Z = 2,592 in4
EXAMPLE 1-014 - 1
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 0
TECHNICAL FEATURES OF SAP2000 TESTED Eigenvalue analysis of a frame with unequal moment of inertia values (I22 I33) for bending modes Automatic frame subdivision RESULTS COMPARISON The independent results are calculated based on formulas presented on page 313 in Clough and Penzien 1975 for a cantilever beam with uniformly distributed mass and constant EI.
Mode
Output Parameter
1
First mode for bending about the Z-axis
Period, sec
2
First mode for bending about the Y-axis
Period, sec
Independent
Percent Difference
Model
SAP2000
A (1 elem)
0.054547
+43.53%
B (2 elems)
0.042333
+11.39%
C (4 elems)
0.039090
+2.85%
D (6 elems)
0.038485
E (8 elems)
0.038273
+0.71%
F (10 elems)
0.038175
+0.45%
G (96 elems)
0.038003
-0.01%
A (1 elem)
0.036364
+43.52%
B (2 elems)
0.028222
+11.39%
C (4 elems)
0.026060
+2.85%
D (6 elems)
0.025657
E (8 elems)
0.025516
+0.71%
F (10 elems)
0.025450
+0.45%
G (96 elems)
0.025335
-0.01%
0.038005
0.025337
+1.26%
+1.26%
EXAMPLE 1-014 - 2
Software Verification PROGRAM NAME: REVISION NO.:
Mode
Output Parameter
Independent
SAP2000 0
Percent Difference
Model
SAP2000
A (1 elem)
N.A.
N.A.
B (2 elems)
0.008218
+35.52%
C (4 elems)
0.006651
+9.68%
D (6 elems)
0.006330
E (8 elems)
0.006214
+2.47%
F (10 elems)
0.006160
+1.58%
G (96 elems)
0.006065
+0.02%
A (1 elem)
N.A.
N.A.
B (2 elems)
0.005479
+35.52%
C (4 elems)
0.004434
+9.67%
D (6 elems)
0.004220
E (8 elems)
0.004143
+2.47%
F (10 elems)
0.004107
+1.58%
G (96 elems)
0.004043
0%
A (1 elem)
N.A.
N.A.
B (2 elems)
N.A.
N.A.
C (4 elems)
0.002511
+15.98%
D (6 elems)
0.002321
E (8 elems)
0.002254
+4.11%
F (10 elems)
0.002222
+2.63%
G (96 elems)
0.002166
+0.05%
3
Second mode for bending about the Z-axis
Period, sec
4
Second mode for bending about the Y-axis
Period, sec
5
Third mode for bending about the Z-axis
Period, sec
0.006064
0.004043
0.002165
+4.39%
+4.38%
+7.21%
EXAMPLE 1-014 - 3
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 0
Note that the SAP2000 results for models A, B, C, D, E, F and G are for lumped mass analyses, with masses lumped 96 inches, 48 inches, 24 inches, 16 inches, 12 inches, 9.6 inches and 1 inch apart, respectively, whereas the independent solution is derived for a uniformly distributed mass. COMPUTER FILES: Example 1-014a, Example 1-014b, Example 1-014c, Example 1-014d, Example 1-014e, Example 1-014f, Example 1-014g CONCLUSION The SAP2000 results show an acceptable match with the independent solution as long as the discretization of the beam is sufficient for the lumped mass analysis to approximate the uniform mass distribution. In this example, the cantilever beam needs to be discretized into a number of elements equal to at least three times the vibration mode considered to obtain acceptable comparison with the independent results. For example, when considering a second mode of vibration (SAP2000 mode numbers 3 and 4 in this example), the beam needs to be discretized into at least 2 * 3 = 6 elements.
EXAMPLE 1-014 - 4
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 0
HAND CALCULATION
EXAMPLE 1-014 - 5
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 0
EXAMPLE 1-014 - 6