SCF Comparison Between Experimental, FEA, and Parametric Equation Results

Stress Concentration Factors (SCF) are the increase in stress at a geometrical discontinuity in a structure such as a hole in a plate or at the intersection of a brace and a chord. In a typical structural analysis, the geometric detail necessary to resolve an SCF is not modeled and is usually not important for a strength analysis. However, for a fatigue analysis, the SCF must be computed and applied to the nominal stress to obtain the peak stress for use with fatigue life calculations. For the tubular structures typically used in offshore structures, there are several parametric equations used to predict the SCF at a brace-chord intersection. The most widely used is the Efthymiou/Durkin parametric equations 1. These equations predict the SCF at the crown and saddle of tubular connections subject to brace axial force and in-plane and out-of-plane bending. The Efthymiou/Durkin parametric equations can be calculated quite easily using an online tool such as 2 or within a structural analysis program. There are other parameteric equations such as the Kuang, Wordsworth/Smedley, Hellier/Connolly/Dover, and Lloyd’s Register equations. We will only address the Efthymiou/Durkin parametric equations.

Outside of the range of applicability of these equations are ring stiffened, grouted, and non-tubular joints for which the SCF is desired but there are no parametric equations from which to compute them. In these cases, the SCF can be readily computed using a Finite Element Analysis (FEA). A number of interpretative problems may arise in the use of FEA in the prediction of the SCF. There may be stress singularities when modeling a joint without modeling the weld. In that case, progressive mesh refinement will eventually yield an infinite SCF.

Likewise, the computation of the SCF for tubular joints can be dependent on the modeling methods used in the FEA model and the method of SCF extraction. The DNVGL Recommended Practice C203 (3) outlines a procedure for extracting the SCF from an FEA. This approach has been codified and automated in a proprietary CRM Engineering program which extracts the SCF along the intersection of an arbitrary connection.

However, little work has been reported in the literature which compares both the parametric and the FEA SCF against experimentally measured SCF. The reference 4 compiles a substantial literature survey of experimental SCF measurements and compares them with the aforementioned parametric equations. Thibaux and Cooreman 5 compare the Efthymiou SCF prediction against an FEA assessment using a rela- tively coarse tetrahedral mesh for the joints.

Approach

In this work we compare the Efthymiou SCF with the experimental results from 4, FEA results from four node shell elements meshed with two different strategies against the results from 4, and the FEA results with the Efthymiou SCF. The FEA models are built from selected joints reported in 4 in which $\beta \lt 1$ and the material is steel for T/Y joints (4, Tables B1.1 - B1.15). There are 37 unique joints in 4 modeled.

Two meshing strategies are used. In the first the $t \times t$ to $2t \times 2t$ method advocated by 3 is employed. This method is indicated by FEA/1. The global mesh seed is $\min\left(2 t_{\mathrm{brace}}, t_{\mathrm{chord}}\right)$. The second method forces the number of elements at the intersection to be 128. This method is indicated by FEA/2. In general, method FEA/2 yields a finer mesh than FEA/1.

FEA Boundary Conditions

At the center of both ends of the chord and at the center of the brace end, a reference point is created. This reference point is tied to the member end with a kinematic coupling constraint. The chord end reference points are fixed. The axial force, in-plane and out-of-plane bending moments are applied to the brace reference point. The magnitudes of the force and moments are calculated to produce a unit stress on the extreme fiber of the brace. Thus the calculated SCF is referenced to unity.

Results

The comparisons are plotted such that a perfect correlation would be on the 1:1 black line in the figures. A curve through the origin is fit to each dataset. If the fitted curve is above the 1:1 line, then the method overestimates the SCF. Likewise, if the fitted curve is below the 1:1 line, then the method underestimates the SCF.

In [68]:
import pandas as pd
import matplotlib
import numpy as np
from scipy import stats
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
matplotlib.style.use('ggplot')

linear_func is the function for a straight line with $b = 0$.

In [69]:
def linear_func(xdata, m):
    return m*xdata

Read in the result summary into a pandas dataframe. Not all the experimental results have results for each loading or location. These missing values are indicated with a NaN.

In [70]:
df = pd.read_csv('result_summary_combined.csv', na_values=['NaN'])
In [71]:
with pd.option_context('display.max_rows', None):
    print(df)
       Joint      Source  AxialChordSaddle  AxialChordCrown  AxialChordSide  \
0        1.1  Experiment               8.3              4.7             8.3   
1        1.1   Efthymiou               7.4              4.4             NaN   
2        1.1         FEA               4.2              3.8             4.9   
3        1.1       FEA/2               4.2              3.8             4.9   
4       1.13  Experiment              13.0              4.8            13.0   
5       1.13   Efthymiou              13.2              4.2             NaN   
6       1.13         FEA               9.4              4.8            10.5   
7       1.13       FEA/2               9.4              4.8            10.5   
8        1.3  Experiment              11.4              5.4            11.4   
9        1.3   Efthymiou              12.1              3.9             NaN   
10       1.3         FEA               8.8              4.5             9.7   
11       1.3       FEA/2               9.6              4.5             9.6   
12       1.5  Experiment              29.0              NaN            29.0   
13       1.5   Efthymiou              15.2              4.3             NaN   
14       1.5         FEA              12.7              6.3            12.9   
15       1.5       FEA/2              11.5              5.9            12.0   
16       1.7  Experiment              10.4              4.7            10.4   
17       1.7   Efthymiou               8.1              4.3             NaN   
18       1.7         FEA               4.7              4.8             5.8   
19       1.7       FEA/2               4.5              4.2             5.1   
20       1.9  Experiment               9.9              2.7             9.9   
21       1.9   Efthymiou               8.8              4.7             NaN   
22       1.9         FEA               6.5              3.6             7.1   
23       1.9       FEA/2               7.3              4.5             7.5   
24     11-12  Experiment               4.7              NaN             4.7   
25     11-12   Efthymiou               4.5              2.2             NaN   
26     11-12         FEA               3.8              1.9             3.9   
27     11-12       FEA/2               4.7              2.1             4.7   
28     13-17  Experiment               7.7              NaN             7.7   
29     13-17   Efthymiou               5.9              2.9             NaN   
30     13-17         FEA               5.0              2.3             5.0   
31     13-17       FEA/2               6.2              2.8             6.2   
32       2-3  Experiment               5.7              NaN             5.7   
33       2-3   Efthymiou               6.8              2.2             NaN   
34       2-3         FEA               5.4              1.8             5.6   
35       2-3       FEA/2               6.5              2.1             6.5   
36      4-10  Experiment               6.7              NaN             6.7   
37      4-10   Efthymiou               7.2              2.2             NaN   
38      4-10         FEA               5.9              1.8             5.9   
39      4-10       FEA/2               6.8              2.0             6.8   
40      SH-A  Experiment              17.3              NaN            17.3   
41      SH-A   Efthymiou              15.8              4.2             NaN   
42      SH-A         FEA              12.1              3.1            12.3   
43      SH-A       FEA/2              14.6              3.7            14.9   
44      SM-A  Experiment              15.8              NaN            15.8   
45      SM-A   Efthymiou              14.8              4.0             NaN   
46      SM-A         FEA              12.6              3.1            12.6   
47      SM-A       FEA/2              14.2              3.4            14.4   
48      SM-B  Experiment              16.1              NaN            16.1   
49      SM-B   Efthymiou              14.2              3.3             NaN   
50      SM-B         FEA              11.7              3.1            11.7   
51      SM-B       FEA/2              12.6              3.3            12.7   
52      SM-C  Experiment              15.2              NaN            15.2   
53      SM-C   Efthymiou              11.7              3.4             NaN   
54      SM-C         FEA               9.0              3.4             9.1   
55      SM-C       FEA/2               9.8              3.4             9.8   
56      SR-A  Experiment              15.2              NaN            15.2   
57      SR-A   Efthymiou              14.4              3.9             NaN   
58      SR-A         FEA              12.1              3.0            12.1   
59      SR-A       FEA/2              13.4              3.4            13.6   
60       T-O  Experiment              11.6              NaN            11.6   
61       T-O   Efthymiou              10.9              2.3             NaN   
62       T-O         FEA               9.5              2.6             9.6   
63       T-O       FEA/2               9.6              2.6             9.7   
64       T-R  Experiment               7.1              NaN             7.1   
65       T-R   Efthymiou               8.5              2.1             NaN   
66       T-R         FEA               7.3              2.1             7.5   
67       T-R       FEA/2               7.9              2.1             7.9   
68      T-ST  Experiment               8.7              NaN             8.7   
69      T-ST   Efthymiou               9.5              2.9             NaN   
70      T-ST         FEA               6.4              2.9             6.7   
71      T-ST       FEA/2               8.0              3.1             8.0   
72   T-Steel  Experiment              18.5              6.4            18.5   
73   T-Steel   Efthymiou              18.3              4.5             NaN   
74   T-Steel         FEA              14.3              4.8            14.7   
75   T-Steel       FEA/2              15.8              5.1            16.4   
76       T19  Experiment              11.4              5.2            11.4   
77       T19   Efthymiou              12.4              3.8             NaN   
78       T19         FEA               9.4              2.8             9.5   
79       T19       FEA/2              11.4              3.3            11.4   
80       T20  Experiment               6.5              2.8             6.5   
81       T20   Efthymiou               6.9              2.2             NaN   
82       T20         FEA               5.2              1.9             5.3   
83       T20       FEA/2               6.6              2.1             6.6   
84      T201  Experiment               6.0              2.9             6.0   
85      T201   Efthymiou               5.4              1.9             NaN   
86      T201         FEA               4.3              1.8             4.3   
87      T201       FEA/2               5.1              2.2             5.1   
88      T202  Experiment               6.1              3.2             6.1   
89      T202   Efthymiou               5.4              1.9             NaN   
90      T202         FEA               4.3              1.8             4.3   
91      T202       FEA/2               5.1              2.2             5.1   
92      T203  Experiment               6.1              2.9             6.1   
93      T203   Efthymiou               5.4              1.9             NaN   
94      T203         FEA               4.3              1.8             4.3   
95      T203       FEA/2               5.1              2.2             5.1   
96     T204C  Experiment               5.7              2.8             5.7   
97     T204C   Efthymiou               5.4              1.9             NaN   
98     T204C         FEA               5.1              2.0             5.1   
99     T204C       FEA/2               5.1              2.2             5.1   
100     T205  Experiment               6.3              3.0             6.3   
101     T205   Efthymiou               5.4              1.9             NaN   
102     T205         FEA               4.3              1.8             4.3   
103     T205       FEA/2               5.1              2.2             5.1   
104     T206  Experiment               6.7              3.2             6.7   
105     T206   Efthymiou               5.4              1.9             NaN   
106     T206         FEA               4.3              1.8             4.3   
107     T206       FEA/2               5.1              2.2             5.1   
108     T207  Experiment               6.5              3.6             6.5   
109     T207   Efthymiou               5.4              1.9             NaN   
110     T207         FEA               4.3              1.8             4.3   
111     T207       FEA/2               5.1              2.2             5.1   
112     T208  Experiment               5.6              3.0             5.6   
113     T208   Efthymiou               5.4              1.9             NaN   
114     T208         FEA               4.3              1.8             4.3   
115     T208       FEA/2               5.1              2.2             5.1   
116     T209  Experiment               5.5              2.8             5.5   
117     T209   Efthymiou               5.4              1.9             NaN   
118     T209         FEA               4.3              1.8             4.3   
119     T209       FEA/2               5.1              2.2             5.1   
120     T210  Experiment               5.5              2.9             5.5   
121     T210   Efthymiou               5.4              1.9             NaN   
122     T210         FEA               4.3              1.8             4.3   
123     T210       FEA/2               5.1              2.2             5.1   
124    T211C  Experiment               6.5              3.1             6.5   
125    T211C   Efthymiou               5.4              1.9             NaN   
126    T211C         FEA               4.3              1.8             4.3   
127    T211C       FEA/2               5.1              2.2             5.1   
128     T212  Experiment               6.6              3.0             6.6   
129     T212   Efthymiou               5.4              1.9             NaN   
130     T212         FEA               4.3              1.8             4.3   
131     T212       FEA/2               5.1              2.2             5.1   
132     T213  Experiment               6.7              3.1             6.7   
133     T213   Efthymiou               5.4              1.9             NaN   
134     T213         FEA               4.3              1.8             4.3   
135     T213       FEA/2               5.1              2.2             5.1   
136     T214  Experiment               5.9              2.9             5.9   
137     T214   Efthymiou               5.4              1.9             NaN   
138     T214         FEA               4.3              1.8             4.3   
139     T214       FEA/2               5.1              2.2             5.1   
140     T215  Experiment               5.6              2.9             5.6   
141     T215   Efthymiou               5.4              1.9             NaN   
142     T215         FEA               4.3              1.8             4.3   
143     T215       FEA/2               5.1              2.2             5.1   
144     T216  Experiment               5.1              3.3             5.1   
145     T216   Efthymiou               4.4              1.8             NaN   
146     T216         FEA               3.2              1.6             3.2   
147     T216       FEA/2               4.1              2.0             4.1   
148    T217C  Experiment               5.0              3.3             5.0   
149    T217C   Efthymiou               4.4              1.8             NaN   
150    T217C         FEA               3.2              1.6             3.2   
151    T217C       FEA/2               4.1              2.0             4.1   
152     T218  Experiment               5.1              3.0             5.1   
153     T218   Efthymiou               4.4              1.8             NaN   
154     T218         FEA               3.2              1.6             3.2   
155     T218       FEA/2               4.1              2.0             4.1   
156     T219  Experiment               4.7              2.9             4.7   
157     T219   Efthymiou               4.4              1.8             NaN   
158     T219         FEA               3.2              1.6             3.2   
159     T219       FEA/2               4.1              2.0             4.1   
160     T223  Experiment              24.2              9.0            24.2   
161     T223   Efthymiou              18.0              4.5             NaN   
162     T223         FEA              14.7              5.5            15.0   
163     T223       FEA/2              15.9              5.7            15.9   
164    T23/1  Experiment               NaN              NaN             NaN   
165    T23/1   Efthymiou               4.5              2.3             NaN   
166    T23/1         FEA               4.3              2.0             4.3   
167    T23/1       FEA/2               4.8              2.2             4.8   
168    T23/2  Experiment               NaN              NaN             NaN   
169    T23/2   Efthymiou               4.5              2.3             NaN   
170    T23/2         FEA               4.3              2.0             4.3   
171    T23/2       FEA/2               4.8              2.2             4.8   
172    T23/3  Experiment               NaN              NaN             NaN   
173    T23/3   Efthymiou               4.5              2.3             NaN   
174    T23/3         FEA               4.3              2.0             4.3   
175    T23/3       FEA/2               4.8              2.2             4.8   
176    T24/1  Experiment               NaN              NaN             NaN   
177    T24/1   Efthymiou               3.1              1.7             NaN   
178    T24/1         FEA               3.1              1.6             3.1   
179    T24/1       FEA/2               3.4              1.6             3.4   
180    T24/2  Experiment               NaN              NaN             NaN   
181    T24/2   Efthymiou               3.1              1.7             NaN   
182    T24/2         FEA               3.1              1.6             3.1   
183    T24/2       FEA/2               3.4              1.6             3.4   
184    T24/3  Experiment               NaN              NaN             NaN   
185    T24/3   Efthymiou               3.1              1.7             NaN   
186    T24/3         FEA               3.1              1.6             3.1   
187    T24/3       FEA/2               3.4              1.6             3.4   
188      T39  Experiment               4.2              2.4             4.2   
189      T39   Efthymiou               4.6              2.3             NaN   
190      T39         FEA               5.7              1.8             5.7   
191      T39       FEA/2               4.9              2.2             4.9   
192      T40  Experiment               NaN              NaN             NaN   
193      T40   Efthymiou               3.1              1.6             NaN   
194      T40         FEA               3.1              1.5             3.1   
195      T40       FEA/2               3.3              1.6             3.3   
196      T41  Experiment              12.0              6.2            12.0   
197      T41   Efthymiou              11.6              3.8             NaN   
198      T41         FEA              11.4              4.5            11.4   
199      T41       FEA/2              10.7              4.3            10.8   
200      T42  Experiment               3.3              1.8             3.3   
201      T42   Efthymiou               2.9              1.1             NaN   
202      T42         FEA               2.6              1.3             2.6   
203      T42       FEA/2               2.6              1.3             2.7   
204      T43  Experiment               4.8              2.9             4.8   
205      T43   Efthymiou               4.6              2.6             NaN   
206      T43         FEA               4.1              1.9             4.1   
207      T43       FEA/2               4.9              2.5             4.9   
208      T44  Experiment               2.6              1.7             2.6   
209      T44   Efthymiou               2.3              1.4             NaN   
210      T44         FEA               2.3              1.3             2.3   
211      T44       FEA/2               2.5              1.4             2.5   
212   T703-1  Experiment               NaN              NaN             NaN   
213   T703-1   Efthymiou              12.6              4.5             NaN   
214   T703-1         FEA               9.0              3.4             9.0   
215   T703-1       FEA/2              11.7              4.0            11.7   
216   T703-2  Experiment               NaN              NaN             NaN   
217   T703-2   Efthymiou              12.6              4.5             NaN   
218   T703-2         FEA               9.0              3.4             9.0   
219   T703-2       FEA/2              11.7              4.0            11.7   
220   T703-3  Experiment               NaN              NaN             NaN   
221   T703-3   Efthymiou              12.6              4.5             NaN   
222   T703-3         FEA               9.0              3.4             9.0   
223   T703-3       FEA/2              11.7              4.0            11.7   
224   T704-1  Experiment               NaN              NaN             NaN   
225   T704-1   Efthymiou               7.1              2.7             NaN   
226   T704-1         FEA               5.4              2.2             5.4   
227   T704-1       FEA/2               6.7              2.5             6.7   
228   T704-2  Experiment               NaN              NaN             NaN   
229   T704-2   Efthymiou               7.1              2.7             NaN   
230   T704-2         FEA               5.4              2.2             5.4   
231   T704-2       FEA/2               6.7              2.5             6.7   
232   T704-3  Experiment               NaN              NaN             NaN   
233   T704-3   Efthymiou               7.1              2.7             NaN   
234   T704-3         FEA               5.4              2.2             5.4   
235   T704-3       FEA/2               6.7              2.5             6.7   
236      TG1  Experiment               5.9              3.3             5.9   
237      TG1   Efthymiou               6.5              2.5             NaN   
238      TG1         FEA               4.9              2.2             4.9   
239      TG1       FEA/2               6.0              2.3             6.1   

     AxialBraceSaddle  AxialBraceCrown  AxialBraceSide  OPBChordSaddle  \
0                 NaN              1.7             NaN             7.9   
1                 3.3              1.3             NaN             8.6   
2                 3.9              2.8             5.0             4.9   
3                 3.9              2.8             5.0             4.9   
4                 6.8              2.2             6.8            12.2   
5                 6.9              1.3             NaN            15.3   
6                10.1              2.4            10.1            10.4   
7                10.1              2.4            10.1            10.4   
8                 8.2              NaN             8.2             NaN   
9                 6.7              1.3             NaN            14.2   
10                9.9              2.4             9.9             9.7   
11                9.9              2.4            10.0            10.5   
12                NaN              1.0             NaN            18.5   
13                8.0              0.5             NaN            18.8   
14               12.5              2.8            12.5            15.6   
15               12.2              2.6            12.2            14.2   
16                6.1              2.4             6.1            10.9   
17                3.5              0.5             NaN            10.0   
18                4.4              2.7             5.9             6.1   
19                4.2              2.4             5.5             5.7   
20                5.0              NaN             5.0             6.2   
21                4.2              2.1             NaN             6.7   
22                6.0              3.7             6.1             5.1   
23                6.3              4.1             6.9             5.8   
24                NaN              NaN             NaN             NaN   
25                5.4              3.0             NaN             2.3   
26                5.4              2.7             5.4             2.1   
27                7.6              3.9             7.6             2.5   
28                NaN              NaN             NaN             NaN   
29                6.0              3.0             NaN             3.0   
30                5.7              2.5             5.7             2.6   
31                8.7              3.8             8.7             3.1   
32                NaN              NaN             NaN             NaN   
33                6.5              2.4             NaN             5.2   
34                6.6              1.7             6.7             4.2   
35                8.5              2.8             8.6             5.0   
36                NaN              NaN             NaN             NaN   
37                6.9              2.3             NaN             5.5   
38                6.8              1.6             6.9             4.6   
39                8.9              2.7             8.9             5.3   
40                NaN              NaN             NaN             NaN   
41               10.4              2.2             NaN            10.5   
42               12.3              1.9            12.3             8.1   
43               14.6              2.4            14.6             9.7   
44                NaN              NaN             NaN             NaN   
45               10.1              2.2             NaN             9.8   
46               11.9              2.1            12.2             8.4   
47               14.4              2.6            14.4             9.5   
48                NaN              NaN             NaN             NaN   
49                9.7              1.8             NaN            12.4   
50               12.0              1.9            12.0             9.9   
51               13.3              2.0            13.4            10.7   
52                NaN              NaN             NaN             NaN   
53                7.9              1.6             NaN            12.9   
54               11.5              2.0            11.6             9.2   
55               11.5              2.0            11.5             9.9   
56                NaN              NaN             NaN             NaN   
57               10.0              2.2             NaN             9.6   
58               11.7              2.1            12.0             8.2   
59               14.1              2.6            14.2             9.0   
60                8.4              NaN             8.4             NaN   
61                9.9              1.4             NaN             9.9   
62               11.1              1.6            11.2             8.7   
63               11.1              1.6            11.6             8.8   
64                6.2              NaN             6.2             NaN   
65                8.1              1.9             NaN             7.4   
66                8.9              1.9             8.9             6.4   
67                9.6              2.0             9.6             6.8   
68                6.0              NaN             6.0             9.1   
69                6.4              1.9             NaN             9.7   
70                8.1              2.0             8.1             6.4   
71                9.0              2.3             9.0             7.7   
72               12.0              NaN            12.0             NaN   
73                9.9              1.5             NaN            16.7   
74               12.6              1.9            12.6            12.7   
75               13.9              2.2            13.9            14.0   
76                6.5              0.3             6.5             NaN   
77                8.3              2.3             NaN             9.3   
78                9.4              1.7             9.4             6.8   
79               11.4              2.1            11.4             8.1   
80                4.9              0.4             4.9             NaN   
81                6.5              2.3             NaN             5.5   
82                7.0              2.0             7.0             4.3   
83                8.5              2.7             8.6             5.3   
84                NaN              NaN             NaN             NaN   
85                4.8              2.2             NaN             5.0   
86                5.3              2.3             5.3             4.1   
87                6.5              2.9             6.5             4.8   
88                NaN              NaN             NaN             NaN   
89                4.8              2.2             NaN             5.0   
90                5.3              2.3             5.3             4.1   
91                6.5              2.9             6.5             4.8   
92                NaN              NaN             NaN             NaN   
93                4.8              2.2             NaN             5.0   
94                5.3              2.3             5.3             4.1   
95                6.5              2.9             6.5             4.8   
96                5.7              1.6             5.7             4.4   
97                4.8              2.2             NaN             5.0   
98                6.4              2.4             6.4             4.7   
99                6.5              2.9             6.5             4.8   
100               6.4              2.7             6.4             NaN   
101               4.8              2.2             NaN             5.0   
102               5.3              2.3             5.3             4.1   
103               6.5              2.9             6.5             4.8   
104               5.3              1.7             5.3             NaN   
105               4.8              2.2             NaN             5.0   
106               5.3              2.3             5.3             4.1   
107               6.5              2.9             6.5             4.8   
108               5.6              2.6             5.6             NaN   
109               4.8              2.2             NaN             5.0   
110               5.3              2.3             5.3             4.1   
111               6.5              2.9             6.5             4.8   
112               5.5              2.5             5.5             NaN   
113               4.8              2.2             NaN             5.0   
114               5.3              2.3             5.3             4.1   
115               6.5              2.9             6.5             4.8   
116               5.9              2.2             5.9             NaN   
117               4.8              2.2             NaN             5.0   
118               5.3              2.3             5.3             4.1   
119               6.5              2.9             6.5             4.8   
120               6.7              2.3             6.7             NaN   
121               4.8              2.2             NaN             5.0   
122               5.3              2.3             5.3             4.1   
123               6.5              2.9             6.5             4.8   
124               5.8              2.0             5.8             NaN   
125               4.8              2.2             NaN             5.0   
126               5.3              2.3             5.3             4.1   
127               6.5              2.9             6.5             4.8   
128               5.8              2.0             5.8             NaN   
129               4.8              2.2             NaN             5.0   
130               5.3              2.3             5.3             4.1   
131               6.5              2.9             6.5             4.8   
132               5.9              2.0             5.9             NaN   
133               4.8              2.2             NaN             5.0   
134               5.3              2.3             5.3             4.1   
135               6.5              2.9             6.5             4.8   
136               6.1              2.2             6.1             NaN   
137               4.8              2.2             NaN             5.0   
138               5.3              2.3             5.3             4.1   
139               6.5              2.9             6.5             4.8   
140               4.9              1.8             4.9             NaN   
141               4.8              2.2             NaN             5.0   
142               5.3              2.3             5.3             4.1   
143               6.5              2.9             6.5             4.8   
144               4.8              2.1             4.8             NaN   
145               4.1              2.3             NaN             4.2   
146               4.4              2.2             4.4             3.3   
147               5.6              2.9             5.7             4.0   
148               5.6              2.4             5.6             NaN   
149               4.1              2.3             NaN             4.2   
150               4.4              2.2             4.4             3.3   
151               5.6              2.9             5.7             4.0   
152               5.2              2.3             5.2             NaN   
153               4.1              2.3             NaN             4.2   
154               4.4              2.2             4.4             3.3   
155               5.6              2.9             5.7             4.0   
156               5.0              2.1             5.0             NaN   
157               4.1              2.3             NaN             4.2   
158               4.4              2.2             4.4             3.3   
159               5.6              2.9             5.7             4.0   
160              14.0              2.2            14.0            22.0   
161               9.4              1.2             NaN            18.6   
162              12.9              2.3            12.9            14.6   
163              13.2              2.5            13.2            15.8   
164               NaN              NaN             NaN             2.2   
165               5.6              3.0             NaN             2.4   
166               5.8              2.7             5.8             2.2   
167               7.8              4.0             7.8             2.5   
168               NaN              NaN             NaN             2.8   
169               5.6              3.0             NaN             2.4   
170               5.8              2.7             5.8             2.2   
171               7.8              4.0             7.8             2.5   
172               NaN              NaN             NaN             2.5   
173               5.6              3.0             NaN             2.4   
174               5.8              2.7             5.8             2.2   
175               7.8              4.0             7.8             2.5   
176               NaN              NaN             NaN             1.8   
177               5.0              3.0             NaN             1.7   
178               4.8              2.7             4.8             1.8   
179               6.2              3.7             6.2             1.9   
180               NaN              NaN             NaN             1.6   
181               5.0              3.0             NaN             1.7   
182               4.8              2.7             4.8             1.8   
183               6.2              3.7             6.2             1.9   
184               NaN              NaN             NaN             1.7   
185               5.0              3.0             NaN             1.7   
186               4.8              2.7             4.8             1.8   
187               6.2              3.7             6.2             1.9   
188               3.0              0.8             4.4             NaN   
189               5.5              3.0             NaN             2.4   
190               5.7              2.6             5.7             2.3   
191               7.8              4.0             7.8             2.6   
192               4.2              1.9             4.4             NaN   
193               4.8              3.0             NaN             1.7   
194               4.7              2.7             4.7             1.8   
195               6.0              3.7             6.0             1.9   
196               7.6              1.2             7.6             NaN   
197               6.5              2.0             NaN            10.1   
198               9.2              2.6             9.2             9.3   
199               9.0              2.5             9.0             8.8   
200               4.6              2.1             4.6             NaN   
201               3.8              2.3             NaN             2.8   
202               3.8              2.2             3.8             2.7   
203               4.2              2.3             4.2             2.8   
204               3.7              0.8             3.7             NaN   
205               4.7              3.0             NaN             2.7   
206               5.2              2.6             5.2             2.4   
207               7.3              4.0             7.3             2.8   
208               3.6              1.8             3.7             NaN   
209               3.7              3.0             NaN             1.4   
210               3.9              2.6             3.9             1.5   
211               4.9              3.4             4.9             1.6   
212               NaN              NaN             NaN             8.0   
213               8.8              2.6             NaN             9.4   
214              10.4              1.7            10.4             6.3   
215              11.9              2.1            11.9             8.3   
216               NaN              NaN             NaN             7.7   
217               8.8              2.6             NaN             9.4   
218              10.4              1.7            10.4             6.3   
219              11.9              2.1            11.9             8.3   
220               NaN              NaN             NaN             8.4   
221               8.8              2.6             NaN             9.4   
222              10.4              1.7            10.4             6.3   
223              11.9              2.1            11.9             8.3   
224               NaN              NaN             NaN             5.4   
225               7.0              2.5             NaN             5.6   
226               7.4              2.0             7.4             4.2   
227               9.1              2.7             9.1             5.3   
228               NaN              NaN             NaN             5.0   
229               7.0              2.5             NaN             5.6   
230               7.4              2.0             7.4             4.2   
231               9.1              2.7             9.1             5.3   
232               NaN              NaN             NaN             5.4   
233               7.0              2.5             NaN             5.6   
234               7.4              2.0             7.4             4.2   
235               9.1              2.7             9.1             5.3   
236               6.3              1.9             6.3             4.9   
237               6.4              2.5             NaN             4.9   
238               6.6              2.1             6.6             3.9   
239               8.4              2.8             8.4             4.6   

     OPBChordSide  OPBBraceSaddle  OPBBraceSide  IPBChordCrown  IPBChordSide  \
0             7.9             4.0           4.0            NaN           NaN   
1             NaN             4.7           NaN            3.8           NaN   
2             5.7             4.7           5.9            2.8           3.2   
3             5.7             4.7           5.9            2.8           3.2   
4            12.2             6.5           6.5            3.9           3.9   
5             NaN             8.2           NaN            4.9           NaN   
6            11.1            10.7          10.7            3.7           3.8   
7            11.1            10.7          10.7            3.7           3.8   
8             NaN             7.3           7.3            4.6           4.6   
9             NaN             7.9           NaN            4.5           NaN   
10           10.4            10.7          10.7            3.5           3.5   
11           10.5            10.7          10.8            3.5           3.6   
12           18.5            10.6          10.6            6.7           6.7   
13            NaN            10.4           NaN            5.5           NaN   
14           15.8            15.0          15.0            4.5           4.5   
15           14.7            14.6          14.6            4.1           4.1   
16           10.9             6.2           6.2            3.3           3.3   
17            NaN             5.7           NaN            4.1           NaN   
18            7.5             5.7           7.8            3.3           3.5   
19            6.7             5.4           7.4            2.8           3.2   
20            6.2             3.6           3.6            3.1           3.1   
21            NaN             4.8           NaN            3.9           NaN   
22            5.2             4.7           4.7            2.8           2.9   
23            5.8             5.0           5.2            3.1           3.2   
24            NaN             NaN           NaN            NaN           NaN   
25            NaN             3.0           NaN            1.5           NaN   
26            2.1             2.6           2.6            1.3           1.3   
27            2.5             3.6           3.6            1.5           1.5   
28            NaN             NaN           NaN            NaN           NaN   
29            NaN             3.3           NaN            1.8           NaN   
30            2.6             2.6           2.7            1.7           1.7   
31            3.1             3.9           3.9            1.9           1.9   
32            NaN             NaN           NaN            NaN           NaN   
33            NaN             5.0           NaN            2.5           NaN   
34            4.2             4.9           5.0            1.6           1.7   
35            5.0             6.4           6.4            1.9           1.9   
36            NaN             NaN           NaN            NaN           NaN   
37            NaN             5.4           NaN            2.3           NaN   
38            4.6             5.2           5.2            1.6           1.8   
39            5.3             6.8           6.8            1.9           2.0   
40            NaN             NaN           NaN            NaN           NaN   
41            NaN             8.1           NaN            4.4           NaN   
42            8.1             8.2           8.2            3.1           3.2   
43            9.8             9.7           9.7            3.7           3.8   
44            NaN             NaN           NaN            NaN           NaN   
45            NaN             7.8           NaN            4.1           NaN   
46            8.4             8.0           8.1            3.0           3.1   
47            9.5             9.6           9.6            3.4           3.6   
48            NaN             NaN           NaN            NaN           NaN   
49            NaN             9.0           NaN            4.0           NaN   
50            9.9            10.3          10.3            2.7           3.1   
51           10.7            11.5          11.5            3.1           3.3   
52            NaN             NaN           NaN            NaN           NaN   
53            NaN             8.3           NaN            3.6           NaN   
54            9.3            11.3          11.3            2.8           2.9   
55            9.9            11.2          11.2            2.8           2.8   
56            NaN             NaN           NaN            NaN           NaN   
57            NaN             7.7           NaN            4.0           NaN   
58            8.2             7.9           8.0            2.9           3.1   
59            9.1             9.5           9.6            3.4           3.5   
60            NaN             NaN           NaN            NaN           NaN   
61            NaN             9.4           NaN            3.2           NaN   
62            8.8            10.3          10.4            2.3           2.5   
63            8.9            10.3          10.7            2.3           2.5   
64            NaN             NaN           NaN            NaN           NaN   
65            NaN             7.1           NaN            2.6           NaN   
66            6.5             7.8           7.8            1.9           2.0   
67            6.8             8.5           8.5            2.0           2.1   
68            9.1             5.1           5.1            3.1           3.1   
69            NaN             6.5           NaN            3.3           NaN   
70            6.6             7.7           7.7            2.5           2.5   
71            7.7             8.5           8.5            2.7           2.7   
72            NaN             NaN           NaN            NaN           NaN   
73            NaN            10.8           NaN            5.9           NaN   
74           13.1            11.7          11.7            4.1           4.5   
75           14.5            12.8          12.8            4.4           4.9   
76            NaN             NaN           NaN            2.5           2.5   
77            NaN             6.7           NaN            3.6           NaN   
78            6.9             6.6           6.6            2.5           2.6   
79            8.1             8.0           8.0            3.0           3.1   
80            NaN             NaN           NaN            1.7           1.7   
81            NaN             5.2           NaN            2.3           NaN   
82            4.4             5.4           5.4            1.7           1.8   
83            5.3             6.6           6.6            1.9           2.0   
84            NaN             NaN           NaN            NaN           NaN   
85            NaN             4.9           NaN            2.3           NaN   
86            4.1             5.1           5.1            1.6           1.8   
87            4.8             6.2           6.2            2.0           2.0   
88            NaN             NaN           NaN            NaN           NaN   
89            NaN             4.9           NaN            2.3           NaN   
90            4.1             5.1           5.1            1.6           1.8   
91            4.8             6.2           6.2            2.0           2.0   
92            NaN             NaN           NaN            NaN           NaN   
93            NaN             4.9           NaN            2.3           NaN   
94            4.1             5.1           5.1            1.6           1.8   
95            4.8             6.2           6.2            2.0           2.0   
96            4.4             4.5           4.5            2.0           2.0   
97            NaN             5.0           NaN            2.3           NaN   
98            4.7             5.9           5.9            1.9           2.0   
99            4.8             6.2           6.2            2.0           2.0   
100           NaN             NaN           NaN            NaN           NaN   
101           NaN             4.9           NaN            2.3           NaN   
102           4.1             5.1           5.1            1.6           1.8   
103           4.8             6.2           6.2            2.0           2.0   
104           NaN             NaN           NaN            NaN           NaN   
105           NaN             4.9           NaN            2.3           NaN   
106           4.1             5.1           5.1            1.6           1.8   
107           4.8             6.2           6.2            2.0           2.0   
108           NaN             NaN           NaN            NaN           NaN   
109           NaN             4.9           NaN            2.3           NaN   
110           4.1             5.1           5.1            1.6           1.8   
111           4.8             6.2           6.2            2.0           2.0   
112           NaN             NaN           NaN            NaN           NaN   
113           NaN             4.9           NaN            2.3           NaN   
114           4.1             5.1           5.1            1.6           1.8   
115           4.8             6.2           6.2            2.0           2.0   
116           NaN             NaN           NaN            NaN           NaN   
117           NaN             4.9           NaN            2.3           NaN   
118           4.1             5.1           5.1            1.6           1.8   
119           4.8             6.2           6.2            2.0           2.0   
120           NaN             NaN           NaN            NaN           NaN   
121           NaN             4.9           NaN            2.3           NaN   
122           4.1             5.1           5.1            1.6           1.8   
123           4.8             6.2           6.2            2.0           2.0   
124           NaN             NaN           NaN            NaN           NaN   
125           NaN             4.9           NaN            2.3           NaN   
126           4.1             5.1           5.1            1.6           1.8   
127           4.8             6.2           6.2            2.0           2.0   
128           NaN             NaN           NaN            NaN           NaN   
129           NaN             4.9           NaN            2.3           NaN   
130           4.1             5.1           5.1            1.6           1.8   
131           4.8             6.2           6.2            2.0           2.0   
132           NaN             NaN           NaN            NaN           NaN   
133           NaN             4.9           NaN            2.3           NaN   
134           4.1             5.1           5.1            1.6           1.8   
135           4.8             6.2           6.2            2.0           2.0   
136           NaN             NaN           NaN            NaN           NaN   
137           NaN             4.9           NaN            2.3           NaN   
138           4.1             5.1           5.1            1.6           1.8   
139           4.8             6.2           6.2            2.0           2.0   
140           NaN             NaN           NaN            NaN           NaN   
141           NaN             4.9           NaN            2.3           NaN   
142           4.1             5.1           5.1            1.6           1.8   
143           4.8             6.2           6.2            2.0           2.0   
144           NaN             NaN           NaN            NaN           NaN   
145           NaN             4.1           NaN            2.1           NaN   
146           3.3             4.1           4.1            1.3           1.5   
147           4.0             5.3           5.3            1.8           1.8   
148           NaN             NaN           NaN            2.0           2.0   
149           NaN             4.1           NaN            2.1           NaN   
150           3.3             4.1           4.1            1.3           1.5   
151           4.0             5.3           5.3            1.8           1.8   
152           NaN             NaN           NaN            NaN           NaN   
153           NaN             4.1           NaN            2.1           NaN   
154           3.3             4.1           4.1            1.3           1.5   
155           4.0             5.3           5.3            1.8           1.8   
156           NaN             NaN           NaN            NaN           NaN   
157           NaN             4.1           NaN            2.1           NaN   
158           3.3             4.1           4.1            1.3           1.5   
159           4.0             5.3           5.3            1.8           1.8   
160          22.0            13.4          13.4            5.9           6.3   
161           NaN            11.9           NaN            6.6           NaN   
162          14.7            13.6          13.6            4.4           5.1   
163          15.8            13.9          13.9            4.7           5.4   
164           2.2             2.0           2.0            NaN           NaN   
165           NaN             3.0           NaN            1.5           NaN   
166           2.2             2.8           2.8            1.4           1.4   
167           2.5             3.6           3.6            1.5           1.5   
168           2.8             2.3           2.3            NaN           NaN   
169           NaN             3.0           NaN            1.5           NaN   
170           2.2             2.8           2.8            1.4           1.4   
171           2.5             3.6           3.6            1.5           1.5   
172           2.5             1.9           1.9            NaN           NaN   
173           NaN             3.0           NaN            1.5           NaN   
174           2.2             2.8           2.8            1.4           1.4   
175           2.5             3.6           3.6            1.5           1.5   
176           1.8             1.7           1.7            NaN           NaN   
177           NaN             2.6           NaN            1.1           NaN   
178           1.8             2.5           2.5            1.1           1.1   
179           1.9             3.1           3.1            1.1           1.1   
180           1.6             1.7           1.7            NaN           NaN   
181           NaN             2.6           NaN            1.1           NaN   
182           1.8             2.5           2.5            1.1           1.1   
183           1.9             3.1           3.1            1.1           1.1   
184           1.7             1.8           1.8            NaN           NaN   
185           NaN             2.6           NaN            1.1           NaN   
186           1.8             2.5           2.5            1.1           1.1   
187           1.9             3.1           3.1            1.1           1.1   
188           NaN             NaN           NaN            0.8           1.0   
189           NaN             3.0           NaN            1.5           NaN   
190           2.3             2.8           2.8            1.3           1.4   
191           2.6             3.7           3.7            1.6           1.6   
192           NaN             NaN           NaN            NaN           NaN   
193           NaN             2.6           NaN            1.1           NaN   
194           1.8             2.5           2.5            1.0           1.1   
195           1.9             3.1           3.1            1.1           1.1   
196           NaN             NaN           NaN            4.2           4.9   
197           NaN             6.7           NaN            4.2           NaN   
198           9.3             7.7           7.7            3.8           3.9   
199           8.8             7.6           7.6            3.6           3.7   
200           NaN             NaN           NaN            1.3           1.3   
201           NaN             3.7           NaN            1.4           NaN   
202           2.7             3.9           3.9            1.2           1.2   
203           2.8             4.3           4.3            1.2           1.2   
204           NaN             NaN           NaN            1.5           1.5   
205           NaN             3.1           NaN            1.7           NaN   
206           2.4             2.7           2.7            1.5           1.4   
207           2.8             3.7           3.7            1.8           1.8   
208           NaN             NaN           NaN            1.0           1.0   
209           NaN             2.3           NaN            1.0           NaN   
210           1.5             2.3           2.3            1.0           1.0   
211           1.6             2.9           2.9            1.0           1.0   
212           8.0             NaN           NaN            NaN           NaN   
213           NaN             6.7           NaN            3.6           NaN   
214           6.3             7.1           7.1            2.4           2.6   
215           8.3             8.1           8.1            3.0           3.1   
216           7.7             NaN           NaN            NaN           NaN   
217           NaN             6.7           NaN            3.6           NaN   
218           6.3             7.1           7.1            2.4           2.6   
219           8.3             8.1           8.1            3.0           3.1   
220           8.4             NaN           NaN            NaN           NaN   
221           NaN             6.7           NaN            3.6           NaN   
222           6.3             7.1           7.1            2.4           2.6   
223           8.3             8.1           8.1            3.0           3.1   
224           5.4             NaN           NaN            NaN           NaN   
225           NaN             5.3           NaN            2.3           NaN   
226           4.2             5.5           5.5            1.7           1.7   
227           5.3             6.7           6.7            1.9           2.0   
228           5.0             NaN           NaN            NaN           NaN   
229           NaN             5.3           NaN            2.3           NaN   
230           4.2             5.5           5.5            1.7           1.7   
231           5.3             6.7           6.7            1.9           2.0   
232           5.4             NaN           NaN            NaN           NaN   
233           NaN             5.3           NaN            2.3           NaN   
234           4.2             5.5           5.5            1.7           1.7   
235           5.3             6.7           6.7            1.9           2.0   
236           4.9             5.4           5.4            1.8           1.8   
237           NaN             4.7           NaN            2.1           NaN   
238           3.9             4.6           4.6            1.7           1.7   
239           4.6             5.9           5.9            1.9           1.9   

     IPBBraceCrown  IPBBraceSide  
0              NaN           NaN  
1              3.2           NaN  
2              2.3           2.5  
3              2.3           2.5  
4              NaN           NaN  
5              3.2           NaN  
6              2.4           3.2  
7              2.4           3.2  
8              2.4           2.4  
9              3.2           NaN  
10             2.4           3.0  
11             2.5           3.1  
12             3.9           3.9  
13             3.7           NaN  
14             2.9           3.8  
15             2.6           3.7  
16             2.4           2.4  
17             3.0           NaN  
18             2.5           2.7  
19             2.3           2.5  
20             2.1           2.1  
21             3.6           NaN  
22             2.8           2.8  
23             3.0           3.1  
24             NaN           NaN  
25             2.2           NaN  
26             2.1           2.1  
27             2.7           2.7  
28             NaN           NaN  
29             2.3           NaN  
30             2.1           2.1  
31             2.8           2.9  
32             NaN           NaN  
33             2.5           NaN  
34             2.0           2.4  
35             2.8           2.8  
36             NaN           NaN  
37             2.6           NaN  
38             2.0           2.4  
39             2.9           2.9  
40             NaN           NaN  
41             3.5           NaN  
42             3.1           3.3  
43             3.6           3.9  
44             NaN           NaN  
45             3.4           NaN  
46             3.1           3.4  
47             3.6           3.9  
48             NaN           NaN  
49             3.3           NaN  
50             2.7           3.2  
51             2.9           3.4  
52             NaN           NaN  
53             2.9           NaN  
54             2.4           2.9  
55             2.4           3.0  
56             NaN           NaN  
57             3.4           NaN  
58             3.1           3.3  
59             3.6           3.9  
60             NaN           NaN  
61             3.3           NaN  
62             2.7           2.9  
63             2.7           2.9  
64             NaN           NaN  
65             2.9           NaN  
66             2.5           2.5  
67             2.6           2.8  
68             1.7           2.0  
69             2.8           NaN  
70             2.2           2.6  
71             2.5           2.9  
72             NaN           NaN  
73             4.1           NaN  
74             2.9           4.0  
75             3.2           4.1  
76             1.7           2.1  
77             2.9           NaN  
78             2.5           2.8  
79             2.8           3.1  
80             2.3           2.4  
81             2.5           NaN  
82             2.3           2.3  
83             2.8           2.8  
84             NaN           NaN  
85             2.6           NaN  
86             2.4           2.4  
87             2.9           2.9  
88             NaN           NaN  
89             2.6           NaN  
90             2.4           2.4  
91             2.9           2.9  
92             NaN           NaN  
93             2.6           NaN  
94             2.4           2.4  
95             2.9           2.9  
96             1.9           2.3  
97             2.6           NaN  
98             2.8           2.8  
99             2.9           2.9  
100            NaN           NaN  
101            2.6           NaN  
102            2.4           2.4  
103            2.9           2.9  
104            NaN           NaN  
105            2.6           NaN  
106            2.4           2.4  
107            2.9           2.9  
108            NaN           NaN  
109            2.6           NaN  
110            2.4           2.4  
111            2.9           2.9  
112            NaN           NaN  
113            2.6           NaN  
114            2.4           2.4  
115            2.9           2.9  
116            NaN           NaN  
117            2.6           NaN  
118            2.4           2.4  
119            2.9           2.9  
120            NaN           NaN  
121            2.6           NaN  
122            2.4           2.4  
123            2.9           2.9  
124            NaN           NaN  
125            2.6           NaN  
126            2.4           2.4  
127            2.9           2.9  
128            NaN           NaN  
129            2.6           NaN  
130            2.4           2.4  
131            2.9           2.9  
132            NaN           NaN  
133            2.6           NaN  
134            2.4           2.4  
135            2.9           2.9  
136            NaN           NaN  
137            2.6           NaN  
138            2.4           2.4  
139            2.9           2.9  
140            NaN           NaN  
141            2.6           NaN  
142            2.4           2.4  
143            2.9           2.9  
144            NaN           NaN  
145            2.4           NaN  
146            2.2           2.2  
147            2.8           2.8  
148            2.0           2.0  
149            2.4           NaN  
150            2.2           2.2  
151            2.8           2.8  
152            NaN           NaN  
153            2.4           NaN  
154            2.2           2.2  
155            2.8           2.8  
156            NaN           NaN  
157            2.4           NaN  
158            2.2           2.2  
159            2.8           2.8  
160            2.8           3.6  
161            4.5           NaN  
162            3.2           4.2  
163            3.4           4.4  
164            NaN           NaN  
165            2.2           NaN  
166            2.1           2.1  
167            2.7           2.7  
168            NaN           NaN  
169            2.2           NaN  
170            2.1           2.1  
171            2.7           2.7  
172            NaN           NaN  
173            2.2           NaN  
174            2.1           2.1  
175            2.7           2.7  
176            NaN           NaN  
177            2.1           NaN  
178            2.0           2.0  
179            2.5           2.5  
180            NaN           NaN  
181            2.1           NaN  
182            2.0           2.0  
183            2.5           2.5  
184            NaN           NaN  
185            2.1           NaN  
186            2.0           2.0  
187            2.5           2.5  
188            1.3           1.5  
189            2.2           NaN  
190            2.1           2.1  
191            2.7           2.7  
192            1.9           1.9  
193            2.1           NaN  
194            2.0           2.0  
195            2.5           2.5  
196            1.9           2.6  
197            3.1           NaN  
198            2.9           3.2  
199            2.8           3.2  
200            2.3           2.3  
201            2.3           NaN  
202            2.2           2.2  
203            2.3           2.3  
204            1.6           1.6  
205            2.3           NaN  
206            2.1           1.5  
207            2.8           2.8  
208            1.9           2.0  
209            2.0           NaN  
210            2.0           2.0  
211            2.4           2.4  
212            NaN           NaN  
213            2.9           NaN  
214            2.4           2.7  
215            2.8           3.1  
216            NaN           NaN  
217            2.9           NaN  
218            2.4           2.7  
219            2.8           3.1  
220            NaN           NaN  
221            2.9           NaN  
222            2.4           2.7  
223            2.8           3.1  
224            NaN           NaN  
225            2.5           NaN  
226            2.2           2.3  
227            2.8           2.8  
228            NaN           NaN  
229            2.5           NaN  
230            2.2           2.3  
231            2.8           2.8  
232            NaN           NaN  
233            2.5           NaN  
234            2.2           2.3  
235            2.8           2.8  
236            2.0           2.0  
237            2.4           NaN  
238            2.2           2.2  
239            2.8           2.8  
In [72]:
def make_plot(x1, y1, x2, y2, popt_1, popt_2,ymax, x_axis, y_axis='FEA'):
    fig = plt.figure(figsize=(10, 8))
    ax = fig.add_subplot(111)
    plt.scatter(x1.values,
                y1.values,
                color='r',
                label='FEA/1')
    plt.scatter(x2.values,
                y2.values,
                color='b',
                label='FEA/2')
    plt.plot([0,100],[0,100],color='k')
    plt.plot(x1.values, 
             linear_func(x1.values, *popt_1), 
             'r')
    plt.plot(x2.values, 
             linear_func(x2.values, *popt_2), 
             'b')
    plt.xlabel(x_axis)
    plt.ylabel(y_axis)
    ax.set_aspect('equal')
    ax.set_xlim(0,ymax)
    ax.set_ylim(0,ymax)
    plt.legend()
    return fig
In [73]:
def make_plot_experiment(x1, y1, x2, y2, popt_1, popt_2, ymax):
    fig = make_plot(x1, y1, x2, y2, popt_1, popt_2,ymax, 'Experiment')
    return fig
In [74]:
def make_plot_experiment_efthymiou(x, y1, popt_1, ymax):
    fig = plt.figure(figsize=(10, 8))
    ax = fig.add_subplot(111)
    plt.scatter(x.values,
                y1.values,
                color='r',
                label='Efthymiou')
    plt.plot([0,100],[0,100],color='k')
    plt.plot(x.values, 
             linear_func(x.values, *popt_1), 
             'r')
    plt.xlabel('Experiment')
    plt.ylabel('Efthymiou')
    ax.set_aspect('equal')
    ax.set_xlim(0,ymax)
    ax.set_ylim(0,ymax)
    plt.legend()
    return fig
In [75]:
def make_plot_efthymiou(x, y1, y2, popt_1, popt_2, ymax):
    fig = make_plot(x, y1, y2, popt_1, popt_2,ymax, 'Efthymiou')
    return fig

The outliers_iqr function removes outliers using the inter-quartile method.

In [76]:
def outliers_iqr(ys):
    
    quartile_1, quartile_3 = np.percentile(ys, [25, 75])
    iqr = quartile_3 - quartile_1
    lower_bound = quartile_1 - (iqr * 1.5)
    #print(lower_bound)
    upper_bound = quartile_3 + (iqr * 1.5)
    #print(upper_bound)
    return np.argwhere((ys > upper_bound) | (ys < lower_bound))
In [77]:
def process_case(lc):
    '''
    lc = 'AxialChordSaddle', etc.
    '''
    experiment = df.loc[df['Source'] == 'Experiment'][lc]
    efthy_exp = df.loc[df['Source'] == 'Efthymiou'][lc]
    fea = df.loc[df['Source'] == 'FEA'][lc]
    fea_2 = df.loc[df['Source'] == 'FEA/2'][lc]
    # Verify the lengths are equal
    assert len( set( [len(experiment), 
                      len(efthy_exp),
                      len(fea),
                      len(fea_2)]) ) == 1
    # Create dataframe from columns
    src = {'Experiment':experiment.values,
           'Efthymiou':efthy_exp.values,
           'FEA':fea.values,
           'FEA/2':fea_2.values}
    src_df = pd.DataFrame(data=src)
    #with pd.option_context('display.max_rows', None):
    #    print(src_df)
    
    # Process FEA
    
    # Drop rows with NaN
    fea_df = src_df.dropna(subset=['Experiment','FEA'])
    # Reindex
    fea_df = fea_df.reset_index(drop=True)
        
    # Detect outliers using inter-quartile range method
    indices_fea = outliers_iqr(fea_df['Experiment'].values - 
                               fea_df['FEA'].values)
    
    # Drop outliers
    fea_df = fea_df.drop(indices_fea.flatten())
    # Reindex
    fea_df = fea_df.reset_index(drop=True)

    # Process FEA/2
    
    # Drop rows with NaN
    fea_2_df = src_df.dropna(subset=['Experiment','FEA/2'])
    # Reindex
    fea_2_df = fea_2_df.reset_index(drop=True)
        
    # Detect outliers using inter-quartile range method
    indices_fea_2 = outliers_iqr(fea_2_df['Experiment'].values - 
                                 fea_2_df['FEA/2'].values)
    
    # Drop outliers
    fea_2_df = fea_2_df.drop(indices_fea_2.flatten())
    # Reindex
    fea_2_df = fea_2_df.reset_index(drop=True)

    # Process Efthymiou
    
    # Drop rows with NaN
    efthy_df = src_df.dropna(subset=['Experiment','Efthymiou'])
    # Reindex
    efthy_df = efthy_df.reset_index(drop=True)
        
    # Detect outliers using inter-quartile range method
    indices_efthy = outliers_iqr(efthy_df['Experiment'].values - 
                                 efthy_df['Efthymiou'].values)
    
    # Drop outliers
    efthy_df = efthy_df.drop(indices_efthy.flatten())
    # Reindex
    efthy_df = efthy_df.reset_index(drop=True)
    
    d = {'fea':fea_df,
         'fea_2':fea_2_df,
         'efthy':efthy_df}
    
    return d

Function which fits both the FEA and the Ethymiou parameteric results to a straight line goig through the origin.

In [78]:
def fit(d):
    popt_1, pcov_1 = curve_fit(linear_func, 
                               d['fea']['Experiment'].values, 
                               d['fea']['FEA'].values)
    popt_2, pcov_2 = curve_fit(linear_func, 
                               d['fea_2']['Experiment'].values, 
                               d['fea_2']['FEA/2'].values)
    popt_3, pcov_3 = curve_fit(linear_func, 
                               d['efthy']['Experiment'].values, 
                               d['efthy']['Efthymiou'].values)
    return ((popt_1, popt_2, popt_3),
            (pcov_1, pcov_2, pcov_3))

Axial

Chord

Saddle

In [79]:
d_axial_chord_saddle = process_case('AxialChordSaddle')
In [80]:
popt_axial_chord_saddle, pcov_axial_chord_saddle = fit(d_axial_chord_saddle)
In [81]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_axial_chord_saddle[0][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_saddle[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 0.76
One standard deviation error on the slope: 0.01
In [82]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_axial_chord_saddle[1][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_saddle[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 0.86
One standard deviation error on the slope: 0.01
In [83]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_axial_chord_saddle[2][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_saddle[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 0.94
One standard deviation error on the slope: 0.01
In [84]:
fig = make_plot_experiment(d_axial_chord_saddle['fea']['Experiment'],
                           d_axial_chord_saddle['fea']['FEA'],
                           d_axial_chord_saddle['fea_2']['Experiment'],
                           d_axial_chord_saddle['fea_2']['FEA/2'], 
                           popt_axial_chord_saddle[0], 
                           popt_axial_chord_saddle[1],
                           20)
fig.savefig('AxialChordSaddle_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Chord/Saddle')
Out[84]:
<matplotlib.text.Text at 0x7fe4e4bae208>
In [85]:
fig = make_plot_experiment_efthymiou(d_axial_chord_saddle['efthy']['Experiment'],
                                     d_axial_chord_saddle['efthy']['Efthymiou'], 
                                     popt_axial_chord_saddle[2],
                                     20)
fig.savefig('AxialChordSaddle_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Chord/Saddle')
Out[85]:
<matplotlib.text.Text at 0x7fe4e4ad2278>

Crown

In [86]:
d_axial_chord_crown = process_case('AxialChordCrown')
In [87]:
popt_axial_chord_crown, pcov_axial_chord_crown = fit(d_axial_chord_crown)
In [88]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_axial_chord_crown[0][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_crown[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 0.66
One standard deviation error on the slope: 0.02
In [89]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_axial_chord_crown[1][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_crown[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 0.76
One standard deviation error on the slope: 0.01
In [90]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_axial_chord_crown[2][0]))
perr = np.sqrt(np.diag(pcov_axial_chord_crown[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 0.71
One standard deviation error on the slope: 0.02
In [91]:
fig = make_plot_experiment(d_axial_chord_crown['fea']['Experiment'],
                           d_axial_chord_crown['fea']['FEA'],
                           d_axial_chord_crown['fea_2']['Experiment'],
                           d_axial_chord_crown['fea_2']['FEA/2'], 
                           popt_axial_chord_crown[0], 
                           popt_axial_chord_crown[1],
                           7)
fig.savefig('AxialChordCrown_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Chord/Crown')
Out[91]:
<matplotlib.text.Text at 0x7fe4e4a5e630>
In [92]:
fig = make_plot_experiment_efthymiou(d_axial_chord_crown['efthy']['Experiment'],
                                     d_axial_chord_crown['efthy']['Efthymiou'], 
                                     popt_axial_chord_crown[2],
                                     8)
fig.savefig('AxialChordCrown_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Chord/Crown')
Out[92]:
<matplotlib.text.Text at 0x7fe4e49e39b0>

Brace

Saddle

In [93]:
d_axial_brace_saddle = process_case('AxialBraceSaddle')
In [94]:
popt_axial_brace_saddle, pcov_axial_brace_saddle = fit(d_axial_brace_saddle)
In [95]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_axial_brace_saddle[0][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_saddle[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 1.06
One standard deviation error on the slope: 0.04
In [96]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_axial_brace_saddle[1][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_saddle[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 1.18
One standard deviation error on the slope: 0.04
In [97]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_axial_brace_saddle[2][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_saddle[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 0.91
One standard deviation error on the slope: 0.03
In [98]:
fig = make_plot_experiment(d_axial_brace_saddle['fea']['Experiment'],
                           d_axial_brace_saddle['fea']['FEA'],
                           d_axial_brace_saddle['fea_2']['Experiment'],
                           d_axial_brace_saddle['fea_2']['FEA/2'], 
                           popt_axial_brace_saddle[0], 
                           popt_axial_brace_saddle[1],
                           16)
fig.savefig('AxialBraceSaddle_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Brace/Saddle')
Out[98]:
<matplotlib.text.Text at 0x7fe4e496e8d0>
In [99]:
fig = make_plot_experiment_efthymiou(d_axial_brace_saddle['efthy']['Experiment'],
                                     d_axial_brace_saddle['efthy']['Efthymiou'], 
                                     popt_axial_brace_saddle[2],
                                     14)
fig.savefig('AxialBraceSaddle_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Brace/Saddle')
Out[99]:
<matplotlib.text.Text at 0x7fe4e48f8470>

Crown

In [100]:
d_axial_brace_crown = process_case('AxialBraceCrown')
In [101]:
popt_axial_brace_crown, pcov_axial_brace_crown = fit(d_axial_brace_crown)
In [102]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_axial_brace_crown[0][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_crown[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 1.15
One standard deviation error on the slope: 0.07
In [103]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_axial_brace_crown[1][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_crown[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 1.36
One standard deviation error on the slope: 0.07
In [104]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_axial_brace_crown[2][0]))
perr = np.sqrt(np.diag(pcov_axial_brace_crown[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 1.02
One standard deviation error on the slope: 0.05
In [105]:
fig = make_plot_experiment(d_axial_brace_crown['fea']['Experiment'],
                           d_axial_brace_crown['fea']['FEA'],
                           d_axial_brace_crown['fea_2']['Experiment'],
                           d_axial_brace_crown['fea_2']['FEA/2'], 
                           popt_axial_brace_crown[0], 
                           popt_axial_brace_crown[1],
                           4)
fig.savefig('AxialBraceCrown_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Brace/Crown')
Out[105]:
<matplotlib.text.Text at 0x7fe4e487a198>
In [106]:
fig = make_plot_experiment_efthymiou(d_axial_brace_crown['efthy']['Experiment'],
                                     d_axial_brace_crown['efthy']['Efthymiou'], 
                                     popt_axial_brace_crown[2],
                                     4)
fig.savefig('AxialBraceCrown_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('Axial/Brace/Crown')
Out[106]:
<matplotlib.text.Text at 0x7fe4e4b8cbe0>

OPB

Chord

Saddle

In [107]:
d_opb_chord_saddle = process_case('OPBChordSaddle')
In [108]:
popt_opb_chord_saddle, pcov_opb_chord_saddle = fit(d_opb_chord_saddle)
In [109]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_opb_chord_saddle[0][0]))
perr = np.sqrt(np.diag(pcov_opb_chord_saddle[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 0.78
One standard deviation error on the slope: 0.02
In [110]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_opb_chord_saddle[1][0]))
perr = np.sqrt(np.diag(pcov_opb_chord_saddle[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 0.92
One standard deviation error on the slope: 0.03
In [111]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_opb_chord_saddle[2][0]))
perr = np.sqrt(np.diag(pcov_opb_chord_saddle[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 1.06
One standard deviation error on the slope: 0.01
In [112]:
fig = make_plot_experiment(d_opb_chord_saddle['fea']['Experiment'],
                           d_opb_chord_saddle['fea']['FEA'],
                           d_opb_chord_saddle['fea_2']['Experiment'],
                           d_opb_chord_saddle['fea_2']['FEA/2'], 
                           popt_opb_chord_saddle[0], 
                           popt_opb_chord_saddle[1],
                           20)
fig.savefig('OPBChordSaddle_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('OPB/Chord/Saddle')
Out[112]:
<matplotlib.text.Text at 0x7fe4e4762940>
In [113]:
fig = make_plot_experiment_efthymiou(d_opb_chord_saddle['efthy']['Experiment'],
                                     d_opb_chord_saddle['efthy']['Efthymiou'], 
                                     popt_opb_chord_saddle[2],
                                     25)
fig.savefig('OPBChordSaddle_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('OPB/Chord/Saddle')
Out[113]:
<matplotlib.text.Text at 0x7fe4e46ece10>

Brace

Saddle

In [114]:
d_opb_brace_saddle = process_case('OPBBraceSaddle')
In [115]:
popt_opb_brace_saddle, pcov_opb_brace_saddle = fit(d_opb_brace_saddle)
In [116]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_opb_brace_saddle[0][0]))
perr = np.sqrt(np.diag(pcov_opb_brace_saddle[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 1.09
One standard deviation error on the slope: 0.05
In [117]:
print('Slope FEA/2 vs experiment: {:0.2f}'.format(popt_opb_brace_saddle[1][0]))
perr = np.sqrt(np.diag(pcov_opb_brace_saddle[1]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/2 vs experiment: 1.23
One standard deviation error on the slope: 0.07
In [118]:
print('Slope: Efthymiou vs experiment: {:0.2f}'.format(popt_opb_brace_saddle[2][0]))
perr = np.sqrt(np.diag(pcov_opb_brace_saddle[2]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope: Efthymiou vs experiment: 1.09
One standard deviation error on the slope: 0.04
In [119]:
fig = make_plot_experiment(d_opb_brace_saddle['fea']['Experiment'],
                           d_opb_brace_saddle['fea']['FEA'],
                           d_opb_brace_saddle['fea_2']['Experiment'],
                           d_opb_brace_saddle['fea_2']['FEA/2'], 
                           popt_opb_brace_saddle[0], 
                           popt_opb_brace_saddle[1],
                           15)
fig.savefig('OPBBraceSaddle_fea_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('FEA vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('OPB/Brace/Saddle')
Out[119]:
<matplotlib.text.Text at 0x7fe4e4666c50>
In [120]:
fig = make_plot_experiment_efthymiou(d_opb_brace_saddle['efthy']['Experiment'],
                                     d_opb_brace_saddle['efthy']['Efthymiou'], 
                                     popt_opb_brace_saddle[2],
                                     12)
fig.savefig('OPBBraceSaddle_efthy_vs_exp.pdf', bbox_inches='tight')
fig.suptitle('Efthymiou vs. Experimental SCF', fontsize=14, fontweight='bold')
fig.axes[0].set_title('OPB/Brace/Saddle')
Out[120]:
<matplotlib.text.Text at 0x7fe4e45f6b70>

IPB

Chord

Crown

In [121]:
d_ipb_chord_crown = process_case('IPBChordCrown')
In [122]:
popt_ipb_chord_crown, pcov_ipb_chord_crown = fit(d_ipb_chord_crown)
In [123]:
print('Slope FEA/1 vs experiment: {:0.2f}'.format(popt_ipb_chord_crown[0][0]))
perr = np.sqrt(np.diag(pcov_ipb_chord_crown[0]))[0]
print('One standard deviation error on the slope: {:0.2f}'.format(perr))
Slope FEA/1 vs experiment: 0.86
One standard deviation error on the slope: 0.03
In [124]:
print('Slope FEA/2 vs experiment: {:0.2f}&