Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 1
EXAMPLE 1-017 FRAME – VIBRATION OF A STRING UNDER TENSION EXAMPLE DESCRIPTION A flexible string is attached to two supports and tensioned to 0.5 kips. The first three modes of the tensioned string are calculated and compared with independent results derived using formulas presented in Kreyszig 1983. The string is modeled using a single frame object. Two different models are created for the comparison. The first has the frame object discretized into 10 elements and the second has it discretized into 100 elements. The initial tension force is applied in the SAP2000 models through a static nonlinear analysis that is specified to consider P-Delta effects. The modal analysis to obtain the frequencies is performed in a second load case. The modal load case is specified to use the stiffness at the end of the static nonlinear case. GEOMETRY, PROPERTIES AND LOADING 0.5 k Z 100" X
Material Properties E = 30,000 k/in2 Mass per unit volume = 7.324E-7 k-sec2/in4 Section Properties 1/16" diameter wire A = 0.00306796 in Frame Object Discretization Model A: 10 frame elements per frame object Model B: 100 frame elements per frame object
TECHNICAL FEATURES OF SAP2000 TESTED Static nonlinear analysis using the P-Delta option to provide tension stiffening Modal analysis of frame for eigenvalues
EXAMPLE 1-017 - 1
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 1
RESULTS COMPARISON The independent results are calculated using string vibration theory presented on pages 506 through 510 of Kreyszig 1983.
Model
A
B
# Elm
10
100
Output Parameter
SAP2000
Independent
Percent Difference
f1 Hz
74.579
74.586
-0.01%
f2 Hz
148.93
149.17
-0.16%
f3 Hz
222.06
223.76
-0.76%
f1 Hz
74.587
74.586
0%
f2 Hz
149.18
149.17
+0.01%
f3 Hz
223.80
223.76
+0.02%
COMPUTER FILES: Example 1-017a, Example 1-017b CONCLUSION The SAP2000 results show an acceptable match with the independent results. Note that the SAP2000 analysis lumps the mass as the element nodes whereas the independent analysis assumes a uniformly distributed mass. As the discretization (number of elements) increases the lumped mass distribution approaches the uniform mass distribution and the SAP2000 results converge to the independent results.
EXAMPLE 1-017 - 2
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 1
HAND CALCULATION
EXAMPLE 1-017 - 3