Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 ASOLID – THICK-WALLED CYLINDER PROBLEM DESCRIPTION In this example, a thick-walled cylinder under plane strain conditions is subjected to a 1 ksi internal pressure, as suggested in MacNeal and Harder 1985. A 10degree segment of the cylinder, 1 inch thick, with a 3-inch inner radius and a 9inch outer radius, is modeled using a 5x1x1 mesh. The radial displacement and the radial, tangential and longitudinal stress components at the inside face of the cylinder are compared with hand-calculated values. Note that radial displacement values are also published in MacNeal and Harder 1985. Six different models using four different Poisson’s ratios and two different meshes are considered in this example. The following table lists the models used. Model
Poisson’s Ratio
Mesh
A
0.3
5x1x1
B
0.49
5x1x1
C
0.499
5x1x1
D
0.4999
5x1x1
E
0.49
20 x 1 x 1
F
0.49
160 x 1 x 1
The incompatible bending modes option is used in all models. The Ux and Uy degrees of freedom are active for the analysis. All other degrees of freedom are inactive.
EXAMPLE 4-002 - 1
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
GEOMETRY, PROPERTIES AND LOADING
Radius
9 .0 0
6.75
5 .2 0
Sym
10°
4 .2 0
Section Properties Inner radius = 3 in Outer radius = 9 in Thickness = 1 in
Free
3 .0 0 3 .5 0
Material Properties E = 1,000 k/in2 ν = 0.49, 0.499, 0.4999
Sym
TECHNICAL FEATURES OF SAP2000 TESTED Analysis using asolid elements Asolid surface pressure load RESULTS COMPARISON Formulas for the independent results are derived from Article 44, entitled Thermal Stresses in a Long, Hollow Cylinder in Timoshenko 1956. The independent results for the radial displacement for Poisson’s ratios of 0.49, 0.499 and 0.4999 are also presented in MacNeal and Harder 1985. The hand-calculated radial displacements based on formulas derived from Timoshenko 1956 are the same as those given in MacNeal and Harder 1985.
EXAMPLE 4-002 - 2
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
With Incompatible Modes Output Parameter
Model, Poisson’s Ratio and Mesh
SAP2000
Independent
Percent Difference*
A, = 0.3, 5x1x1
0.004538
0.004582
-1%
B, = 0.49, 5x1x1
0.004971
0.005040
-1%
Radial displ in
C, = 0.499, 5x1x1
0.004989
0.005060
-1%
At inner face
D, = 0.4999, 5x1x1
0.004991
0.005062
-1%
E, = 0.49, 20x1x1
0.005035
0.005040
0%
F, = 0.49, 160x1x1
0.005040
0.005040
0%
A, = 0.3, 5x1x1
-0.834
-1.000
-17% (-13%)
B, = 0.49, 5x1x1
-0.833
-1.000
-17% (-13%)
Radial stress ksi
C, = 0.499, 5x1x1
-0.833
-1.000
-17% (-13%)
At inner face
D, = 0.4999, 5x1x1
-0.833
-1.000
-17% (-13%)
E, = 0.49, 20x1x1
-0.954
-1.000
-5% (-4%)
F, = 0.49, 160x1x1
-0.994
-1.000
-1% (-0%)
A, = 0.3, 5x1x1
1.293
1.250
3%
B, = 0.49, 5x1x1
1.364
1.250
9%
C, = 0.499, 5x1x1
1.369
1.250
10%
D, = 0.4999, 5x1x1
1.369
1.250
10%
E, = 0.49, 20x1x1
1.291
1.250
3%
F, = 0.49, 160x1x1
1.256
1.250
0%
A, = 0.3, 5x1x1
0.138
0.075
84% (5%)
B, = 0.49, 5x1x1
0.260
0.123
111% (11%)
C, = 0.499, 5x1x1
0.267
0.125
114% (11%)
D, = 0.4999, 5x1x1
0.268
0.125
114% (11%)
E, = 0.49, 20x1x1
0.165
0.123
34% (3%)
F, = 0.49, 160x1x1
0.128
0.123
4% (0%)
Tangential stress ksi At inner face
Longitudinal stress ksi At inner face
* Percent differences in parenthesis are relative to the maximum theoretical tangential stress, 1.250 ksi
EXAMPLE 4-002 - 3
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
COMPUTER FILES: Example 4-002a, Example 4-002b, Example 4-002c Example 4-002d, Example 4-002e, Example 4-002f CONCLUSIONS The SAP2000 results show an acceptable comparison with the independent results when an adequate mesh is used. As the mesh is refined, the SAP2000 results converge to the theoretical results.
EXAMPLE 4-002 - 4
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
HAND CALCULATION
EXAMPLE 4-002 - 5
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 6
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 7
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 8
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 9
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 10
Software Verification PROGRAM NAME: REVISION NO.:
SAP2000 2
EXAMPLE 4-002 - 11