HOVGAARD PROBLEM
1HOVGAARD PROBLEM
Problem definition
Figure 2: ROHR2 MODEL
Figure 1: NUREG MODEL
For the above system, verify the eigenfrequencies, mode shapes and deflections due to a given excitation.
References (NUREG)
P.Bezler/ M. Hartzman/ M. Reich, Piping Benchmark Problems, Vol. 1,
Dynamic Analysis Uniform Support Motion; Response Spectrum Method,
Division of Technical Information and Document Control, U.S. Nuclear Regulatory Commission, Washington D.C., 1980, Chapter 3.1, pg. 15; Chapter 4.1, pg. 2347
This problem is a simple, threedimensional piping system made up of only straight line and bend pipe elements between two fixed anchors (node 1; 11) see figure Figure 1.
The computation of bend element flexibility factors assumed that there is no internal pressure. The system has no distributed mass (no internal generation of element mass). All masses are input as lumped masses. For the solution only five frequencies were calculated. A single input spectrum, shown in Figure , is applied with the weighting factors' of 1.0, 0.667 and 1.0 in the X, Y and Z global directions respectively. As all resultant system natural frequencies are spaced greater than 10% apart, clustering does not occur and the solution is therefore independent of the combination sequence employed. This problem was selected as a benchmark because its simplicity allows the checking of key results.
Model data:
Variable 
Description 
Unit 
Used Value 
Modulus of Elasticity 
lbs/inchÂ² 
24000000 

Poisson's ratio 
 
0.3 

Outside Diameter 
inch 
7.288 

Wall thickness 
inch 
0.241 

Bend radius 
inch 
36.3 

Mass of several nodes 
lb 
 
Table 1: Overview of the used variables
The system is submitted to the following response spectrum:
Figure 3: Response spectrum
Model description (ROHR2)
The coordinates of the nodes in the EPIPE input file are given in inch. They are converted to feet for the ROHR2 input:
Table 2: NODAL POINT NUMBER
The masses of the nodes in the EPIPE input file are given in slinch. They are converted to pound for the ROHR2 input:
A load case â€œEigenvalues was created to calculate the first 5 mode shapes.
In addition a load case â€œdynamic earthquakeâ€ was generated to apply the requested response spectra. The superposition was set to SRSS for the mode shapes.
Result comparisons
The following five frequencies are listed in the resultsoutput of the EPIPEdocumentation.
Value 
No 
Reference NUREG [1/sec] 
ROHR2 [1/sec] 
Difference [%] 
1 
28.535 
28.530 
<0.02 

2 
55.772 
55.811 
<0.07 

3 
81.500 
81.414 
<0.11 

4 
141.742 
141.755 
<0.01 

5 
162.820 
163.236 
<0.26 
Table 1: Comparison of the resulting frequencies
Table 2: Eigenfrequencies
Deflections of the first Eigenvalue
The figures an tables below compare only two points of each mode shape. All results are given in the global coordinate system. The defections of the mode shapes can be scaled to any arbitrary value. In the reference calculation the scaling was based on a generalized mass of 1.0 slinch inÂ².
In the ROHR2 model the generalized mass is automatically chosen as 0.009369586 slinch inÂ².
Value 
Æ’ [1/sec] 
Reference NUREG mass= 1 slinch inÂ² [inch] 
ROHR2 mass= 9,36E3 slinch inÂ² [inch] 
ROHR2 gmass= 1 slinch inÂ² [inch] 
Difference [%] 
28.530 
1.38 
0.13 
1.34 
<3 

0.37 
0.04 
0.41 
<11 

1.94 
0.19 
1.96 
<2 
Table 3: Comparison of the deflection of node 6
Deflections of the second Eigenvalue
Value 
Æ’ [1/sec] 
Reference NUREG mass= 1 slinch inÂ² [inch] 
ROHR2 mass= 9,36E3 slinch inÂ² [inch] 
ROHR2 mass =1 slinch inÂ² [inch] 
Difference [%] 
55.811 
2.28 
0.24 
2.27 
<1 

0.01 
0.01 
0.10 
< 

1.57 
0.15 
0.155 
<2 
Deflections of the third Eigenvalue
Value 
Æ’ [1/sec] 
Reference NUREG mass= 1 slinch inÂ² [inch] 
ROHR2 mass= 9,36E3 slinch inÂ² [inch] 
ROHR2 mass =1 slinch inÂ² [inch] 
Difference [%] 
81.415 
1.21 
0.12 
1.24 
<3 

0.02 
0.00 
0.00 
 

0.62 
0.06 
0.62 
<1 
Deflections of the fourth Eigenvalue
Value 
Æ’ [1/sec] 
Reference NUREG mass= 1 slinch inÂ² [inch] 
ROHR2 mass= 9,36E3 slinch inÂ² [inch] 
ROHR2 mass =1 slinch inÂ² [inch] 
Difference [%] 
141.755 
0.42 
0.04 
0.41 
<3 

0.75 
0.07 
0.72 
<4 

2.02 
0.20 
2.01 
<1 
Deflection of the fifth Eigenvalue
Value 
Æ’ [1/sec] 
Reference NUREG mass= 1 slinch inÂ² [inch] 
ROHR2 mass= 9,36E3 slinch inÂ² [inch] 
ROHR2 mass =1 slinch inÂ² [inch] 
Difference [%] 
163.236 
1.51 
0.15 
1.55 
<3 

1.73 
0.17 
1.76 
<2 

0.28 
0.03 
0.31 
<11 
Participation Factors
The figures an tables below compare the Participation Factors. All results are given in the global coordinate system. The Participation Factors of the mode shapes can be scaled to any arbitrary value. In the reference calculation the scaling was based on a generalized mass of 1.0 slinch inÂ².
In the ROHR2 model the generalized mass is automatically chosen as 0.009369586 slinch inÂ².
Value 
Æ’ [1/sec] 
Reference NUREG gmass= 1 slinch inÂ² [slinch in] 
ROHR2 gmass= 9,36E3 slinch inÂ² [slinch in] 
ROHR2 gmass= 1 slinch inÂ² [slinch in] 
Difference [%] 
SMWX 
28.530 
0.175 
0.017 
0.175 
<1 
SMWY 
0.025 
0.002 
0.025 
<1 

SMWZ 
0.331 
0.032 
0.331 
<1 
Value 
Æ’ [1/sec] 
Reference NUREG gmass= 1 slinch inÂ² [slinch in] 
ROHR2 gmass= 9,36E3 slinch inÂ² [slinch in] 
ROHR2 gmass= 1 slinch inÂ² [slinch in] 
Difference [%] 
SMWX 
81.415 
0.053 
0.005 
0.054 
<2 
SMWY 
0.258 
0.025 
0.258 
<1 

SMWZ 
0.028 
0.002 
0.028 
<1 
Node Displacements
(direction factors 1000, 666, 1000)
Point 4
Value 
Reference NUREG [in] 
ROHR2 [in] 
ROHR2 [in] 
Difference [%] 
WXa 
7.41E3 
7.52 
7.52E3 
<2 
WYa 
6.12E4 
0.62 
6.20E4 
<2 
WZa 
1.74E2 
17.69 
1.77E2 
<2 
Point 8
Value 
Reference NUREG [in] 
ROHR2 [in] 
ROHR2 [in] 
Difference [%] 
WXa 
5.82E3 
5.90 
5.90E3 
<2 
WYa 
1.99E3 
2.02 
2.02E3 
<2 
WZa 
1.68E3 
1.70 
1.70E3 
<2 
Conclusion
The results are generally close to the references by the NUREGexample (benchmark problems 1, Hovgaard problem). There are several reasons for minor differences.
The first one results of the scale factor in ROHR2, which is different to EPIPE. The referenceprogram used a unitymatrix to calculate the deflections of each node. This matrix has a constant value of 1 tmÂ². In ROHR2 it is used a alternative method scale the deflections. It calculates a generalized mass, which generate displacement which are easy to interpret. The following value was used:
In order to compare the deflections between NUREG and ROHR2, it is necessary to multiply the EPIPEresults by the generalized mass of mode shape. The second inaccuracy of the results is shown at point 4 (Load case Eigenvalue; second Eigenvalue; deflection Î´fY). If the result is relatively small (Î´fY= 0,0059 inch), the relative difference will become bigger, though the difference of the displacement vector length and direction is actually negligible.
Files
Problem1_0.r2w
SIGMA Ingenieurgesellschaft mbH www.rohr2.com