CC BY
5doi: 10.5862/MCE.63.4
Surrogate modeling for initial rotational stiffness of welded tubular joints
Суррогатное моделирование для определения начальной жесткости вращения сварных трубчатых соединений
M.R. Garifullin, A.V. Barabash, E.A. Naumova, O.V. Zhuvak,
Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia
T. Jokinen, M. Heinisuo,
Tampere University of Technology, Tampere, Finland
Аспирант М.Р. Гарифуллин, студент А.В. Барабаш, студент Е.А. Наумова, студент О.В. Жувак,
Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Россия
Научный сотрудник Т. Йокинен, PhD, профессор М. Хейнисуо,
Технологический университет Тампере, Тампере, Финляндия
Key words: surrogate modeling; kriging; square hollow section; plane bending; finite element analysis
Ключевые слова: суррогатное моделирование; кригинг; квадратный полый профиль; плоскость изгиба; анализ методом конечных элементов
Abstract. Recently, buildings and structures erected in Russia and abroad have to comply with stringent economic requirements. Buildings should not only be reliable and safe, have a beautiful architectural design, but also meet the criteria of rationality and energy efficiency. In practice, this usually means the need for additional comparative analysis in order to determine the optimal solution to the engineering task. Usually such an analysis is time-consuming and requires huge computational efforts. In this regard, surrogate modeling can be an effective tool for solving such problems. This article provides a brief description of surrogate models and the basic techniques of their construction, describes the construction process of a surrogate model to calculate initial rotational stiffness of welded RHS joints made of high strength steel (HSS).
Аннотация. В последнее время сильно возросли экономические требования, предъявляемые к зданиям и сооружениям, возводимым в России и за границей. Здания должны быть не только надежными и безопасными, иметь красивый внешний вид, но также удовлетворять критериям рациональности и энергоэффективности. На практике это означает необходимость дополнительного сравнительного анализа для определения оптимального решения инженерной задачи. Обычно такой анализ трудоемок и требует огромных вычислительных затрат. В связи с этим суррогатное моделирование может быть эффективным инструментом для решения таких проблем. В данной статье приводится краткое описание суррогатных моделей и основных методов их построения, описывается процесс строительства суррогатной модели для расчета начальной жесткости вращения сварных соединений квадратного полого профиля из высокопрочной стали.
Introduction
Tubular structures with welded joints are used in a wide range of structural applications. The most typical application is tubular truss. The structural analysis model is frequently constructed using beam finite elements, and the braces are connected to the chords using hinges. Actually, a welded joint does not behave as a hinge when it is loaded by a moment. The joint has resistance against the moment, but in the joint area deformations may occur both at the brace and at the chord, so the stiffness against the moment has to be taken into account in the global analysis of the structure. In [1] only the moment
resistance is given for the joint where the angle between the brace and the chord is 90 degrees. In [2] there is the equation, which can be used to calculate the initial rotational stiffness for the same case, angle 90 degrees.
When aiming to economic and environmental friendly design stiffness of the joints must be taken into account. This is especially true when using high strength steel in structures, because then buckling at the ultimate limit state and deflections and vibrations in the serviceability limit state are often critical. Besides, the impact of the installation process and operation of the construction on its strength properties should be considered [3-8]. In [9] and [10] it has been shown that the rotational stiffness of the welded tubular joint is the main parameter when considering buckling of members of tubular trusses.
In the design it is possible to define the rotational stiffness for the joint using comprehensive finite element analysis (FEA). In practice, this is impossible, especially when performing optimization of structures when the structural analysis must be done thousands of times. In order to avoid computationally heavy calculations, surrogate models (or meta models) have been developed. The surrogate model is the basis of a new direction in the simulation engineering. It is a mathematical method of drawing up a model based on the test results and/or computational experiments carried out with a variety of objects of the class in question [11]. Surrogate models have been used widely in aerospace [12-16], civil engineering [17]. Methods of using surrogate models for optimizing steel structures are described in [19-27]. The optimum design of steel frames is presented using semi-rigid joints and surrogate models [18]. This article shows construction of the surrogate model in the case of initial rotational stiffness of welded tubular structures.
This article describes the construction process of a surrogate model for calculating initial rotational stiffness of welded RHS joints made of high strength steel (HSS). Only joints with butt welds are considered. This assumption was made to simplify the surrogate and finite element modeling. The effect of fillet welds is considered in [28].
In the surrogate model construction, we replace the computationally expensive function f(x) with a sum of two other functions, where (x) is the vector of the variables [16]), which has the same dimension of input and output parameters as the original function [29, 30]:
where s(x) is the surrogate model at the point x and e(x) is the difference between the two.
The idea is to use the function s(x) during calculations or optimization instead of the function f(x). The function s(x) is chosen so that it should be cheap to evaluate, and hereby the computation time can be reduced considerably.
We can start with a quadratic regression model:
or with linear regression (fi = 0) or with the constant, only /30 £ 0. If this gives good results (see later criteria), we can add to the regression a predictor Z(x) (stochastic process) and end up to Kriging.
Kriging is the most popular method for creating surrogate models. In [31-48] it was made well-known in the context of modeling, and optimization of deterministic functions, respectively.
The Kriging models consist of two components. The first component is some simple model that captures the trend in the data, and the second component measures deviation between the simple model and the true function. An example of the surrogate model f (x) using Kriging with one variable x with n sample points is:
Theoretical background
Surrogate model construction
f (x ) = s(x ) + s(x) ,
(1)
(2)
__1 n
f (x) = - Z X; + Z (x) ,
(3)
where the zero order regression is used and the predicted value f(x) is given scaled to [0;1].
The real values f(x) can be calculated from the normalized data f(x). In the construction of Z(x) we need a correlation function between points. Define R as the matrix R of stochastic-process correlations between the sample points x, and x/.
Rj = R(0, Xi, Xj ) i, j, = 1,..., n (4)
and let r (x) be a vector of correlation between sample points and untried points x:
r (x) = [r(0, x1, x)... R(0, xn, x )f (5)
The mostly preferred correlation function is the Gaussian correlation:
R(xi, xj )= exP
m
xk - xk
' J
-Z3
k=1
(6)
where Qk are unknown correlation parameters, k = 1,..., m; m is the number of design variables;
k k
x. and x are the components of samples xi and x.
After this the surrogate model can be defined, see e.g. [16].
The article [29] indicates the main problems which arise in the construction of surrogate models and their optimization. For instance, the task of reducing the dimension [49], the task of building a multidimensional nonlinear approximating dependencies problem of clustering and classifying data.
The surrogate model can be used with data of both low and high accuracy. These data of low accuracy can be obtained by the analytical method and the data of high accuracy - during the "field" test [50].
The obtained approximate dependence is necessary not only to predict the result, but also to determine the accuracy of the calculation [51]. One of the basic principles to ensure greater accuracy of calculations in constructing the surrogate model is the removal of a set of superfluous or redundant parameters [52]. In particular, the task of reducing the dimension [53] not only greatly simplifies the calculations, but also allows the surrogate model to meet the required conditions [54]. In addition, the method of reducing the dimension of predicates (of explanatory parameters) is practiced [55].
For certain areas, where it is necessary to build a model, which gives the most accurate information, it is necessary to build several approximating dependencies. Thus, the final approximation procedure will include a classifier that determines which private approximator needs to be taken for the given input variable [30].
Some special cases of surrogate model construction are also considered in [56-61].
Surrogate model validation
The validation process uses a new sample size approximately equal to one third of the sample size used to build the surrogate model [62]. The validation process consists of comparing the results of the surrogate model with those of the real response. This is a specific problem which depends on the accuracy required of the fitted model. If this accuracy is too low, the surrogate model must be modified by the introduction of more sample points or by the modification of the surrogate model variables.
The validation process consists of testing a new set of sample points, but excluding the original sample point set. The accuracy of the surrogate model can be checked using R2 value [18]. No single rule exists that specifies a minimum R2 value which guarantees a good fitting surrogate model. In [18]
2 2 only the surrogate models with R values larger than 0.85 are considered. We realized that R was not
proper in our case; all our models had R2 > 0.95, so we had to refuse to consider it.
For validation we applied the relative error, Eq. (0.1). Usually, the optimization procedure for tubular trusses requires no more than 10% average relative error, so it is accepted as our main criterion.
Error =
C - C
\ FEM ^SURR]
c
^ Z7Z7A Ä
(7)
Design of Experiment
The method for determining the sample points to carry out an analysis is called the Design of Experiments (DOE). The location of the sample points is very important for generating an accurate surrogate model. It consists of a compromise between the usage of a reasonable number of sample points to build an accurate model. Several DOE methods are described in [63-65]. The Latin Hypercube Sampling (LHS) proposed by [63] is the most popular space filling sampling technique. In this research engineering justification is used to define the sample points.
Variables
At the first step of sampling the list of variables should be determined. Every Y joint can be described completely by the following variables (Fig. 1): chord dimensions b0, t0; brace dimensions b1, t{, angle < between the brace and the chord; material properties fy0, fy1.
Figure 1. Dimensions of Y joint
The number of variables has a strong effect on the process of surrogate modeling: the lesser the number of variables is, the easier it is to create a reasonable surrogate model, so it was reasonable to reduce the number of variables.
First of all, we realized that the influence of t1 on rotational stiffness was rather weak and excluded it from the surrogate modeling. Besides, in this research we considered joints with only butt welds. In the case of butt welds material properties have no effect on the initial rotational stiffness of joints, so we also excluded fy0 and fy1 from the list of variables.
We also replaced b1 with its relative analogy fi=b1/b0 so that this variable had similar values for all chords. It should be noted that due to low values of fi, compared to other variables, it was important to input fi with at least four characters after the decimal point to avoid the loss of precision.
After all, we presented the function of the initial rotational stiffness as a function of four variables:
C = f (b0,t0, fi, <) (8)
Sampling
At the next step sample points should be determined. The sample points should be defined so that:
• they cover the whole range of our interest;
• they meet requirements of building codes
• the failure of the brace is not critical.
We considered cold-formed tubes with discrete sizes followed those of Ruukki. Only square sections (SHS) were considered. Our goal was to predict typical practical cases with the chord sizes b0 between 100x100x4 and 300x300x12.5. We considered HSS up to S700 and this also limited the range of cross-sections.
In this research the Eurocodes were used for joints [1] and extension for steel grades up to S700 [66]. The main requirement restricted the range of sections: 0.25 < p < 0.85.
The ratio bi/ti was limited by bi/ti < 35 and to cross-section class 1 or 2. The ratio b0/t0 was limited by 10 < bo/to < 35 and, moreover, to the cross-section class 1 or 2.
The angle q> between the brace and the chord is due to welding in the range of 30 degrees < q> < 90 degrees.
Moreover, we chose the sample points in such a way that for every variable there were 3 different values (minimum, middle and maximum), while others remained constant. Exceptions were made only for cases with the maximum t0 and p, because they had complicated modeling in Abaqus. For those cases only two values (minimum and middle) of t0were considered.
Fig. 2 presents the distribution of sample points in respect to chord dimensions. Black crosses are the possible sections of Ruukki catalogue; blue points are the chosen sample points. Eurocode limits are marked with green lines. Similarly, Fig. 3 presents the relative distributions of sample points in respect to brace dimensions.
Figure 2. Sample points: b0-t0
Figure 3. Sample points: p-b0
As a whole, we chose 285 sample points. To calculate the values of their initial rotational stiffness, the comprehensive FEA was exploited.
Finite element analysis
To calculate values of rotational stiffness in sample points, we conducted the Finite Element Analysis (FEA) in Abaqus. The model was made using C3D8 brick elements. All sections were modeled with round corners, according to [67]. Two-layered mesh was created with solid hexahedral elements, being refined near the joints, as shown in Fig. 4. The butt welds were modeled as "no weld" by using TIE constraints (Fig. 4).
Figure 4. FE model for Y-joint
The material does not influence the stiffness of joints with butt welds, so we applied the elastic material with the modulus of elasticity 210 GPa and Poisson's ratio 0.3.
The analyses were force controlled, and the load step was calculated with "Static, General" procedure. The joint rotation C was calculated from FEA by extracting the frame behavior from the FEA results, as given in [68].
The FEA models were validated with the tests of LUT [69] in [68]. The verification was done in three steps [70]: moment load in two opposite directions, use of shell elements instead of brick ones and varying the type of brick elements from 8 to 20 nodes. The proposed FEA model seemed to work well and was used for surrogate modeling.
Finally, the list of sample points is presented in Table 1.
Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M. Surrogate modeling for initial rotational stiffness of welded tubular joints. Magazine of Civil Engineering. 2016. No. 3. Pp. 53-76. doi: 10.5862/MCE.63.4 58
Table 1. Sample points
о 0 3 о 0 6 о 0 CO о 0 3 о 0 6 о 0 9 о 0 3 о 0 6 о 0 CO
s^
bo ß ti to C [kNm/rad] to C [kNm/rad] to C [kNm/rad]
100 0.400 4 4 55 27 23 6 174 85 72 10 1082 406 345
100 0.600 4 4 215 86 68 6 634 262 211 10 4007 1229 1013
100 0.800 4 4 1135 442 343 6 2847 1107 891
110 0.364 4 4 44 23 20 5 83 43 37 6 140 72 62
110 0.545 4 4 150 63 50 5 272 116 94 6 450 193 158
110 0.818 4 4 1457 568 439 5 2349 948 751 6 3536 1389 1117
120 0.333 4 5 70 37 33 7.1 203 106 92 10 638 291 253
120 0.583 4 5 364 150 121 7.1 1009 422 345 10 3197 1155 953
120 0.833 5 5 2923 1170 923 7.1 6637 2532 2047
140 0.286 4 5 53 30 27 7.1 152 85 76 10 453 231 205
140 0.571 5 5 353 143 117 7.1 944 399 328 10 2646 1075 891
140 0.786 5 5 2097 794 618 7.1 4846 1954 1569
150 0.267 4 6 81 47 42 8.8 262 146 130 12.5 1004 433 382
150 0.533 5 6 448 191 158 8.8 1372 593 494 12.5 5046 1897 1586
150 0.800 6 6 3785 1471 1149 8.8 9697 3742 3034
160 0.250 4 6 73 44 39 8.8 236 135 122 12.5 862 397 353
160 0.563 5 6 559 232 190 8.8 1677 714 590 12.5 5884 2242 1866
160 0.750 6 6 2493 943 735 8.8 6531 2617 2111
180 0.278 4 7.1 148 85 76 8.8 280 158 141 12.5 946 467 415
180 0.556 6 7.1 896 378 309 8.8 1617 693 571 12.5 5165 2134 1774
180 0.833 7.1 7.1 8286 3312 2600 8.8 13374 5401 4322
200 0.250 4 7.1 123 74 66 8.8 233 138 124 12.5 750 402 360
200 0.550 5 7.1 869 367 295 8.8 1566 673 546 12.5 4910 2040 1672
200 0.800 7.1 7.1 6690 2589 2020 8.8 10763 4362 3457
220 0.273 4 8 203 118 105 10 395 226 203 12.5 840 454 405
220 0.545 7.1 8 1157 490 402 10 2131 922 762 12.5 4347 1899 1580
220 0.818 7.1 8 11157 4415 3474 10 18372 7443 5944
250 0.280 4 8.8 285 164 146 10 416 237 212 12.5 859 472 422
250 0.560 7.1 8.8 1718 721 589 10 2429 1034 848 12.5 4815 2094 1731
250 0.800 7.1 8.8 12549 4943 3876 10 16880 6768 5384
260 0.269 4 8.8 267 155 139 10 389 226 203 12.5 798 448 402
260 0.538 7.1 8.8 1496 639 524 10 2118 916 755 12.5 4175 1842 1527
260 0.846 8.8 8.8 20765 8271 6441 10 26909 10941 8628
300 0.267 5 10 390 227 205 12.5 774 445 401
300 0.533 8 10 2136 908 747 12.5 4114 1795 1486
300 0.833 10 10 27313 10664 8367 12.5 45620 18624 14858
Surrogate model construction
Attempt I
Today there are a number of methods for surrogate modeling. We started the construction of the surrogate model with a linear regression, but the error term R2 was rather low. Next we exploited Kriging, as a surrogate model type to approximate deterministic noise-free data. Firstly, we used the DACE toolbox for Matlab [71] with zero, linear and second order regression [72] but we did not manage to construct a reasonable model. For our next surrogate modeling we exploited the ooDACE toolbox for Matlab (hereinafter - ooDACE) [73-74] and those results are reported in the article.
We constructed surrogate models of two types: single model (one model for all sample points) and multi-model (with an independent model for every b0). The idea of implementing the second approach was that the variable b0 is discrete, getting its values from the Ruukki catalogue, with no intermediate values among them. Both types gave rather close results to each other and were used for our final model. It is worth saying that the multi-model requires much less computational time than a single model approach.
Our first validation gave us the following results: R = 0.8876, average error about 56% and maximum error 678 %. To explore the behavior of the model in detail, we plotted the graphs with rotational stiffness in respect to the different variables. Graphical validation demonstrated that the model behaved very unpredictably (Figs. 5-7).
60 <p [deg]
Figure 5. C-9 response
Figure 6. C-t0 response
Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M. Surrogate modeling for initial rotational stiffness of welded tubular joints. Magazine of Civil Engineering. 2016. No. 3. Pp. 53-76. doi: 10.5862/MCE.63.4 60
Chord 200x8.8
12000
-2000
0,2 0,3 0,4 0,5 0,6 0,7 0,8
a
Figure 7. C-p response
Pseudo points
During the validation of our model we came to conclusion that we needed more sample points to make it work properly. Calculating new sample points in Abaqus represented a complicated task and required time, so we decided to implement new points by other means.
To improve the behavior of the model, we decided to add certain boundary conditions for the model. It is obvious that for the angle close to 90 degrees (87...89 degrees) the C-y curve must have a zero slope ( very close to a horizontal line), see Fig 8. Analytically this means that the partial derivative dC/dy must equal to zero. Practically, to apply this boundary condition to a discrete function, we added some sample points for 95 and 100 degrees angles with the same stiffness value as for 90 degrees. To add the boundary conditions for low angles, we extrapolated stiffness for 20 and 25 degrees using the 4th order polynomial regression. Moreover, we decided to add the points between the existed sample ones. Graphically it is shown in Fig. 8 (for C-y response).
We called these additional points that were determined not by Abaqus, but by other means, as "pseudo" points.
100
80
■u
я
£ 60
о
40
20
15
Chord 100x4. /3=0.4
■ У = 5Е-06х4- С ),0014х3 + 0.1661Х2- 8.7773Х + 203,83
X ' ; ;
H— I ■ ■
ISample points I Ext sample points lint sample points
30
45
75
60 Ф [deg] Figure S. Pseudo points
90
105
After applying pseudo points, we managed to improve the C-q> graph (Fig. 9).
Chord 100x4, brace 40x4
60
50
(0
E 40
z ^
o
30
20
30 45 60 75 90
<p [deg] Figure 9. Improved C-9 curve.
To improve the behavior of the surrogate model in respect to other variables (t0 and P), we introduced additional pseudo points using the idea that for zero values of these variables stiffness responses got zero values as well. Pseudo points were also added for two thicknesses (one between the lowest and the middle and one between the middle and the highest) and three betas (one between the lowest and the middle and two between the middle and the highest).
Overall, we added 1869 pseudo points (both extrapolated and interpolated), resulting with 285 sample points the total number of 2154 points. The whole range of pseudo points is presented in Table 2 (for 100 mm chord only).
Table 2. Pseudo points
№ b0 [mm] t0 [mm] P v [deg] C [kNm/rad]
Pseudo 9 int 100 4 0.40 45 36
Pseudo 9 int 100 4 0.40 75 24
Pseudo 9 ext 100 4 0.40 20 75
Pseudo 9 ext 100 4 0.40 25 64
Pseudo 9 ext 100 4 0.40 95 23
Pseudo 9 ext 100 4 0.40 100 23
Pseudo 9 int 100 6 0.40 45 113
Pseudo 9 int 100 6 0.40 75 74
Pseudo 9 ext 100 6 0.40 20 239
Pseudo 9 ext 100 6 0.40 25 203
Pseudo 9 ext 100 6 0.40 95 72
Pseudo 9 ext 100 6 0.40 100 72
Pseudo 9 int 100 10 0.40 45 602
Pseudo 9 int 100 10 0.40 75 352
Pseudo 9 ext 100 10 0.40 20 1654
Pseudo 9 ext 100 10 0.40 25 1338
Pseudo 9 ext 100 10 0.40 95 345
Pseudo 9 ext 100 10 0.40 100 345
Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M. Surrogate modeling for initial rotational stiffness of welded tubular joints. Magazine of Civil Engineering. 2016. No. 3. Pp. 53-76. doi: 10.5862/MCE.63.4 62
№ bo [mm] to [mm] ß Ф [deg] C [kNm/rad]
Pseudo ф int 100 4 0.60 45 128
Pseudo ф int 100 4 0.60 75 71
Pseudo ф ext 100 4 0.60 20 310
Pseudo ф ext 100 4 0.60 25 258
Pseudo ф ext 100 4 0.60 95 68
Pseudo ф ext 100 4 0.60 100 68
Pseudo ф int 100 6 0.60 45 381
Pseudo ф int 100 6 0.60 75 219
Pseudo ф ext 100 6 0.60 20 911
Pseudo ф ext 100 6 0.60 25 759
Pseudo ф ext 100 6 0.60 95 211
Pseudo ф ext 100 6 0.60 100 211
Pseudo ф int 100 10 0.60 45 2010
Pseudo ф int 100 10 0.60 75 1030
Pseudo ф ext 100 10 0.60 20 6435
Pseudo ф ext 100 10 0.60 25 5087
Pseudo ф ext 100 10 0.60 95 1013
Pseudo ф ext 100 10 0.60 100 1013
Pseudo ф int 100 4 0.80 45 669
Pseudo ф int 100 4 0.80 75 359
Pseudo ф ext 100 4 0.80 20 1637
Pseudo ф ext 100 4 0.80 25 1364
Pseudo ф ext 100 4 0.80 95 343
Pseudo ф ext 100 4 0.80 100 343
Pseudo ф int 100 6 0.80 45 1653
Pseudo ф int 100 6 0.80 75 922
Pseudo ф ext 100 6 0.80 20 4184
Pseudo ф ext 100 6 0.80 25 3451
Pseudo ф ext 100 6 0.80 95 891
Pseudo ф ext 100 6 0.80 100 891
Pseudo t0 int 100 5 0.40 30 100
Pseudo t0 int 100 8 0.40 30 466
Pseudo t0 ext 100 0 0.40 30 0
Pseudo t0 ext 100 0.1 0.40 30 0
Pseudo t0 ext 100 12 0.40 30 2225
Pseudo t0 ext 100 12.5 0.40 30 2622
Pseudo t0 int 100 5 0.60 30 374
Pseudo t0 int 100 8 0.60 30 1694
Pseudo t0 ext 100 0 0.60 30 0
Pseudo t0 ext 100 0.1 0.60 30 0
Pseudo t0 ext 100 12 0.60 30 8418
Pseudo t0 ext 100 12.5 0.60 30 9969
Pseudo t0 int 100 5 0.80 30 1896
Pseudo t0 int 100 8 0.80 30 5324
Pseudo t0 ext 100 0 0.80 30 0
Pseudo t0 ext 100 0.1 0.80 30 0
Pseudo t0 ext 100 10 0.80 30 8564
Pseudo t0 ext 100 12 0.80 30 12570
Pseudo t0 ext 100 12.5 0.80 30 13691
№ bo [mm] to [mm] ß Ф [deg] C [kNm/rad]
Pseudo t0 int 100 5 0.40 60 50
Pseudo t0 int 100 8 0.40 60 201
Pseudo t0 ext 100 0 0.40 60 0
Pseudo t0 ext 100 0.1 0.40 60 0
Pseudo t0 ext 100 12 0.40 60 736
Pseudo t0 ext 100 12.5 0.40 60 843
Pseudo t0 int 100 5 0.60 60 157
Pseudo t0 int 100 8 0.60 60 612
Pseudo t0 ext 100 0 0.60 60 0
Pseudo t0 ext 100 0.1 0.60 60 0
Pseudo t0 ext 100 12 0.60 60 2225
Pseudo t0 ext 100 12.5 0.60 60 2548
Pseudo t0 int 100 5 0.80 60 738
Pseudo t0 int 100 8 0.80 60 2068
Pseudo t0 ext 100 0 0.80 60 0
Pseudo t0 ext 100 0.1 0.80 60 0
Pseudo t0 ext 100 10 0.80 60 3325
Pseudo t0 ext 100 12 0.80 60 4878
Pseudo t0 ext 100 12.5 0.80 60 5312
Pseudo t0 int 100 5 0.40 90 43
Pseudo t0 int 100 8 0.40 90 172
Pseudo t0 ext 100 0 0.40 90 0
Pseudo t0 ext 100 0.1 0.40 90 0
Pseudo t0 ext 100 12 0.40 90 622
Pseudo t0 ext 100 12.5 0.40 90 712
Pseudo t0 int 100 5 0.60 90 126
Pseudo t0 int 100 8 0.60 90 501
Pseudo t0 ext 100 0 0.60 90 0
Pseudo t0 ext 100 0.1 0.60 90 0
Pseudo t0 ext 100 12 0.60 90 1842
Pseudo t0 ext 100 12.5 0.60 90 2112
Pseudo t0 int 100 5 0.80 90 586
Pseudo t0 int 100 8 0.80 90 1690
Pseudo t0 ext 100 0 0.80 90 0
Pseudo t0 ext 100 0.1 0.80 90 0
Pseudo t0 ext 100 10 0.80 90 2740
Pseudo t0 ext 100 12 0.80 90 4041
Pseudo t0 ext 100 12.5 0.80 90 4406
Pseudo ß int 100 4 0.50 30 89
Pseudo ß int 100 4 0.70 30 524
Pseudo ß ext 100 4 0.00 30 0
Pseudo ß ext 100 4 0.01 30 0
Pseudo ß ext 100 4 0.90 30 2194
Pseudo ß ext 100 4 0.95 30 2945
Pseudo ß int 100 6 0.50 30 293
Pseudo ß int 100 6 0.70 30 1397
Pseudo ß ext 100 6 0.00 30 0
Pseudo ß ext 100 6 0.01 30 0
Pseudo ß ext 100 6 0.90 30 5308
Pseudo ß ext 100 6 0.95 30 7035
№ be [mm] to [mm] в ф [deg] C [kNm/rad]
Pseudo p int 100 10 0.50 30 2303
Pseudo p int 100 10 0.70 30 6136
Pseudo p ext 100 10 0.00 30 0
Pseudo p ext 100 10 0.01 30 0
Pseudo p ext 100 10 0.90 30 11102
Pseudo p ext 100 10 0.95 30 12334
Pseudo р int 100 4 0.50 60 39
Pseudo р int 100 4 0.70 60 205
Pseudo р ext 100 4 0.00 60 0
Pseudo p ext 100 4 0.01 60 0
Pseudo p ext 100 4 0.90 60 859
Pseudo p ext 100 4 0.95 60 1156
Pseudo p int 100 6 0.50 60 132
Pseudo p int 100 6 0.70 60 552
Pseudo p ext 100 6 0.00 60 0
Pseudo p ext 100 6 0.01 60 0
Pseudo p ext 100 6 0.90 60 2054
Pseudo p ext 100 6 0.95 60 2721
Pseudo p int 100 10 0.50 60 716
Pseudo p int 100 10 0.70 60 2053
Pseudo p ext 100 10 0.00 60 0
Pseudo p ext 100 10 0.01 60 0
Pseudo p ext 100 10 0.90 60 5207
Pseudo p ext 100 10 0.95 60 6437
Pseudo p int 100 4 0.50 90 32
Pseudo p int 100 4 0.70 90 160
Pseudo p ext 100 4 0.00 90 0
Pseudo p ext 100 4 0.01 90 0
Pseudo p ext 100 4 0.90 90 666
Pseudo p ext 100 4 0.95 90 897
Pseudo p int 100 6 0.50 90 109
Pseudo p int 100 6 0.70 90 444
Pseudo p ext 100 6 0.00 90 0
Pseudo p ext 100 6 0.01 90 0
Pseudo p ext 100 6 0.90 90 1659
Pseudo p ext 100 6 0.95 90 2201
Pseudo p int 100 10 0.50 90 596
Pseudo p int 100 10 0.70 90 1688
Pseudo p ext 100 10 0.00 90 0
Pseudo p ext 100 10 0.01 90 0
Pseudo p ext 100 10 0.90 90 4312
Pseudo p ext 100 10 0.95 90 5344
Attempt II
Using pseudo points for ф, t0 and в, we managed to construct a new surrogate model with the following parameters: R2 = 0.9645, average error 8% and maximum error 28 %,16 points with errors higher than 10% (red points). The same results were observed using a multi-model approach. The graphical validation (Figs. 10, 11) showed that the model behaved properly but its accuracy should be significantly improved.
Figure 10. C-9 response
Figure 11. C-t0 and C-p responses
3D plots in Fig. 12-14 illustrate how the model behaves in respect to several variables simulteneously.
Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M. Surrogate modeling for initial rotational stiffness of welded tubular joints. Magazine of Civil Engineering. 2016. No. 3. Pp. 53-76. doi: 10.5862/MCE.63.4 66
£>0=150 тт, <0=6 тт, /3=0.5...0.8, ^=30...Э0в
4000 3500 -3000-
^ 2500 -2
£ 2000 -■г.
О 1500-
1000 -
Figure 12. ^(ф+Р) response
Ь0=ЮО тт, (0=4...Ю тт, ¿3=0.4...0.8, <р=30°
9000 8000 7000 , 6000 5000
Е
2 4000 О 3000
"О
я
0.80
*0 [тт]
4.0 0.40 Р
Р1диге 13. С-^о+Р) гевропве
i>0=100 mm, f0=4...10 mm, J3=0.4, p=30...90°
1200
1000 -
^ 800 73 2
£ 600 z
° 400 -
200
10.0
tp so 4.0 tg [mm]
Figure 14. C-(t0+<p) response Accuracy improvements
Analyzing the validation points, we came to conclusion that all inaccurate cases were related to the points for which fi was predicted (only fi or together with other variables). As it was mentioned before, when choosing sample points for every case we calculated three betas: minimum (0.25.0.4), middle (0.5.0.55) and maximum (0.8.0.85), let call them fii, fi2 and fi3 respectively. We have noticed that for the points with fii < fi < fi2 the predicted values were lower, whereas for the points with fi2 < fi < fi3 the opposite trend was observed. Graphically it is shown in Fig. 15.
Chord 300x12.5, q>=30°
■o 30000
tU
£ 25000 20000
o
0,25
0,40
0,55
0,70
0,85
0,55
* P
Figure 15. Differences in C-p curves before (left) and after (right) improvements
As it can be seen from the graph, the curve predicted in Excel using three sample points (blue line) did not suit accurately the actual curve obtained by using additional points (red line). This discrepancy led to inaccurate values of pseudo points used later in surrogate modeling and, eventually, caused
considerable errors in predicted values, up to 28% (green rectangular). The same difference was observed for all cases.
To tackle this discrepancy, we modified beta curves by changing pseudo points. It was done manually and had no scientific basis, but it allowed us to receive more accurate curves (Fig. 15).
Having improved p curves, we constructed a new surrogate model. Validation of the models showed that this time the number of red points decreased twice (from 16 to 8), the average error reduced a little (from 8% to 7%). However, for some points extremely high errors were observed (up to 75 %).Validation for a multi-model approach brought slightly different results: 11 red points, 6 % average error and 28 % maximum.
We also noticed a similar difference in C-y curves. For the points with y = 45° the stiffness values were about 6% higher, whereas for the points with y = 745° they were 2% lower. This trend repeated also for the multi-model approach.
To tackle this discrepancy, we applied the same technique as before for p curves and constructed a new surrogate model. We did not manage to receive a working surrogate for the single-model approach, although we did it for the multi-model. The results were rather questionable, although this model predicted the best results for 110 mm chords among the models created before.
Despite the fact that none of improved models met the required criteria of accuracy, they proved that it was possible to modify the model by changing the pseudo points and all these models were used to construct the complex surrogate model.
Final model
For that moment we had 5 surrogate models:
• Two without any improvements (single model and multi-model);
• Two with improved betas (single model and multi-model);
• A multi-model with improved betas and angles.
None of them had satisfactory results. However, their performance was different for various chords. It seemed logical to create a complex model which contained a surrogate model for every chord, which suited it best. Solving this task, every chord was analyzed separately to choose a surrogate model with the best performance. The analysis is given in 0. Here number 1 relates to a model without any improvements, number 2 to the model with improved betas and number 3 to the model with improved betas and angles, SM relates to single model, MM to a multi model.
Table 3. Complex model
bo [mm] 1SM 1MM 2SM 2MM 3MM
100 x
110 x
120 x
140 x
150 x
160 x
180 x
200 x
220 x
250 x
260 x
300 x
All the models were collected combined in the final model. Its validation is presented in 0. The average error was 4%, maximum 16%.
Table 4. Final validation
Chord Brace FEM Surrogate model
b0 t0 ß V C C Error [%]
100 8 0.800 79 1838 1735 5.6
100 6 0.400 42 115 122 6.2
100 10 0.400 80 349 348 0.4
100 8 0.500 89 300 285 5.0
110 5 0.364 34 71 72 1.4
110 6 0.364 55 76 75 1.2
110 6 0.364 50 82 81 0.9
110 5 0.818 88 746 748 0.3
120 5.6 0.583 80 170 171 0.8
120 7.1 0.750 83 1098 1111 1.2
120 8.8 0.833 42 6030 6726 11.6
120 7.1 0.500 39 399 387 3.0
140 7.1 0.357 90 106 113 6.5
140 6 0.786 89 977 1017 4.1
140 7.1 0.500 40 393 400 1.7
140 6 0.500 30 349 366 4.8
150 7.1 0.533 47 405 400 1.2
150 7.1 0.800 89 1643 1783 8.5
150 6 0.400 71 84 77 8.0
150 7.1 0.267 39 104 105 0.9
160 8.8 0.750 86 1946 2102 8.0
160 10 0.313 76 244 243 0.3
160 8.8 0.500 85 416 438 5.3
160 8.8 0.438 81 303 321 6.0
Chord Brace FEM Surrogate model
b0 t0 ß V C C Error [%]
180 10 0.389 59 405 391 3.4
180 12.5 0.667 33 8504 9029 6.2
180 10 0.500 40 1085 1096 1.1
180 10 0.611 60 1415 1470 3.9
200 8.8 0.250 69 130 129 1.0
200 12.5 0.400 62 855 811 5.1
200 8 0.300 57 138 131 4.8
200 7.1 0.600 66 448 518 15.5
220 8.8 0.545 71 569 547 3.9
220 12.5 0.455 58 1195 1247 4.4
220 10 0.409 37 703 819 16.4
220 8 0.727 62 1757 1974 12.4
250 12.5 0.720 60 6397 6163 3.7
250 10 0.600 90 1073 1086 1.2
250 12.5 0.600 87 2199 2170 1.3
250 8.8 0.280 72 151 151 0.2
260 10 0.846 79 8820 8802 0.2
260 12.5 0.577 67 2135 2115 0.9
260 12.5 0.308 73 498 522 4.8
260 12.5 0.692 51 6208 6222 0.2
300 12.5 0.833 33 39163 40996 4.7
300 12.5 0.467 56 1290 1396 8.2
300 12.5 0.833 85 14946 14908 0.3
300 12.5 0.600 86 2247 2338 4.1
This is the best surrogate model we managed to construct. Since it met the requirements of accuracy introduced above, we accepted the model to optimize the procedure. The detailed description of the surrogate modeling process is provided in [75]. The model is available for free download at [76].
Alternative methods
We also tried alternative approaches for constructing the surrogate model.
First a linear regression model [77] using existing Matlab tools was tried. It was found out that the results were not satisfactory. Therefore, the model was restricted to considering only the nearest sampling space to every validation point. It presents a validation point as:
*
x =
[b* 10 ß p ]
(9)
The nearest points were chosen to meet the following conditions:
b* - b
< 30
'0~u0i \Po-Pi\< 0.08
K -k| <30
Then the chosen points were sorted by their normalized distances from the validation point:
(10)
dish =.
4
I
k=1
max xk - xk max Xk - min xk
(11)
2
Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M. Surrogate modeling for initial rotational stiffness of welded tubular joints. Magazine of Civil Engineering. 2016. No. 3. Pp. 53-76. doi: 10.5862/MCE.63.4 70
where x1, x2, x3, x4 relate to b0, t0, fi and y respectively.
Then 6 nearest points were taken to form the local linear model. Exploiting this procedure, we managed to construct a model with the following results: R2 = 0.9552, average error 21 % and maximum error 115 %.
We tried also an approach using the Matlab toolbox called Polyfitn. It constructs a polynomial regression model using traditional linear least squares techniques. Using this toolbox, we managed to construct a model with the following results: R2 = 0.9552, average error 37 % and maximum error 454 %.
We couldt be satisfied with none of these models and had to reject both.
Conclusions
1. There exists no analytical method to calculate the initial rotational stiffness for welded tubular Y joints for different angles y. Surrogate modeling based on the comprehensive FEA might be a reasonable solution to this task. The developed model can be utilized to optimize tubular frames and trusses, as it avoids resorting to time-consuming FE analyses.
2. In this article the Kriging method, realized in ooDACE toolbox for Matlab, was exploited for surrogate modeling. It was shown that the original number of 285 sample points was not enough to construct a physically reasonable surrogate model. To make the model behave properly, the additional sample (pseudo) points were applied without exploiting the comprehensive FEA. Besides, utilizing Latin Hypercube Sampling (LHS), instead of engineering justification, could avoid the problem of sampling.
3. Interpolated pseudo points can cause a considerable loss of accuracy which can be avoided through several iterations. Main attention should be paid to C-fi curves, as the variable fi plays a dominating role in the surrogate model behavior.
4. The idea of a multi-model approach might be very effective in surrogate modeling in case some of the variables are of discrete type. In this article the final surrogate model was constructed from several others which had the best performance for every chord width.
5. The surrogate model is constructed for Y joints with butt welds loaded by an in-plane bending moment. However, its application might be expanded to consider other joints and loadings by taking into account the effect of fillet welds, the effect of axial forces, the effect of residual stresses, etc. Moreover, the similar model can be constructed for other types of joints (N, K, KT joints) which are widely used in tubular trusses.
References
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2. Grotmann D., Sedlacek G. Rotational stiffness of welded RHS beam-to-column joints. Cidect 5BB-8/98. RWTH-Aachen. Aachen. 1998.
3. Al Ali M., The Welding Process as a Local. Issue with Global Consequences. Advanced Materials Research. 2014. Vol. 969. Pp. 340-344.
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6. Vatin N., Havula J., Martikainen L., Sinelnikov A., Orlova A.V., Salamakhin S.V. Thin-walled cross-sections and their joints: Tests and FEM-Modelling. Advanced Materials Research. 2014 Vols. 945-949. Pp. 1211-1215.
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8. Vostrov V.K., Vasilkin A.A. Optimizaciya visot poyasov stenki rezervuara [Optimization of heights of zones of a wall of the tank]. Montazhnye i Spetsial'nye Raboty v Stroitel'stve. 2005. No. 11. Pp. 37-39.
Литература
1. European Committee for Standardisation, (CEN). Eurocode 3. Design of steel structures, Part 1-8: Design of joints (EN 1993-1-8:2005). Brussels, 2005.
2. Grotmann D., Sedlacek G. Rotational stiffness of welded RHS beam-to-column joints. Cidect 5BB-8/98 // RWTH-Aachen. Aachen. 1998.
3. Al Ali M., The Welding Process as a Local. Issue with Global Consequences // Advanced Materials Research. 2014. Vol. 969. Pp. 340-344.
4. Al Ali M., Daneshjo N., Size and Distribution of Welding Stresses // Procedia Engineering. 2012. Vol. 40. Pp. 2-7.
5. Garifullin M., Trubina D., Vatin N. Local buckling of cold-formed steel members with edge stiffened holes // Applied Mechanics and Materials. 2015. Vols. 725-726. Pp. 697702.
6. Vatin N., Havula J., Martikainen L., Sinelnikov A., Orlova A.V., Salamakhin S.V. Thin-walled cross-sections and their joints: Tests and FEM-Modelling // Advanced Materials Research. 2014 Vols. 945-949. Pp. 1211-1215.
7. Vatin N., Sinelnikov A., Garifullin M., Trubina D. Simulation of cold-formed steel beams in global and distortional buckling // Applied Mechanics and Materials. 2014. Vols. 633-634. Pp. 1037-1041.
8. Востров В.К., Василькин А.А. Оптимизация высот поясов стенки резервуара // Монтажные и специальные работы в строительстве. 2005. № 11. Pp. 37-39.
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10. Snijder H.H., Boel H.D., Hoenderkamp J.C.D., Spoorenberg R.C. Buckling length factors for welded lattice girders with hollow section braces and chords. Proceedings of Eurosteel. 2011. Pp. 1881-1886.
11. Prikhodko P.V. Primeneniye metodov agregatsii ekspertov i regressii na osnove gaussovskikh protsessov dlya postroyeniya metamodeley Candidate of physico-mathematical sciences dissertation. Moscow institute of physics and technology. Moskow. 2013. Pp.1-26.
12. Roux W.J., Stander N., Haftka R.T. Response surface approximations for structural optimization. International Journal for Numerical Methods in Engineering. 1998. Vol. 42. No. 3. Pp. 517-534.
13. Jin R., Chen W., Simpson T.W. Comparative studies of metamodelling techniques under multiple modelling criteria. Structural and Multidisciplinary Optimization. 2001. Vol. 23. No. 1. Pp. 1-13.
14. Queipo N.V., Haftka R.T., Shyy W., Goel T., Vaidyanathan R., Kevin Tucker P. Surrogate-based analysis and optimization. Progress in Aerospace Sciences. 2005. Vol. 41. Pp. 1-28.
15. Kleijnen J.P.C. Simulation experiments in practice: statistical design and regression analysis. Journal of Simulation. 2008. Vol. 2. Pp. 19-27.
16. Muller J. Surrogate Model Algorithms for Computationally Expensive Black-Box Global Optimization Problems. Tampere University of Technology. Publication 1092. 2012.
17. Mukhopadhyay T., Dey T.K., Dey S., Chakrabarti A. Optimization of fiber reinforced polymer web core bridge deck - A hybrid approach. Structural Engineering International. 2015. Vol. 25. No. 2. Pp. 173-183.
18. Diaz C., Victoria M., Querin O.M., Marti P. Optimum design of semi-rigid connections using metamodels. Journal of Constructional Steel Research. 2012. Vol. 78. Pp. 97-106.
19. Yun G.J., Ghaboussi J., Elnashai A.S. Self-learning simulation method for inverse nonlinear modeling of cyclic behavior of connections. Computer Methods in Applied Mechanics and Engineering. 2008. Vol. 197. No. 33-40. Pp. 2836-2857.
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31. Matheron G. Principles of geostatistics // Economic geology. 1963. Vol. 58. No. 8. Pp. 1246-1266.
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Marsel Garifullin,
+7(999)0346070; [email protected] Aleksandra Barabash,
+79633129679; [email protected]
Elizaveta Naumova, 89215965574; [email protected]
Oksana Zhuvak,
8(911)0961816; [email protected] Timo Jokinen,
+358401981280; timo.jokinen@tut. fi Markku Heinisuo,
+35804(0)5965826; [email protected]
Марсель Ринатович Гарифуллин, +7(999)0346070; эл. почта: 273marcel@gmail. com
Александра Викторовна Барабаш, +79633129679; эл. почта: [email protected]
Елизавета Алексеевна Наумова, 89215965574; эл. почта: [email protected]
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Тимо Йокинен,
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Маркку Хейнисуо, +35804(0)5965826; эл. почта: markku. heinisuo@tut. fi
© Garifullin M.R., Barabash A.V., Naumova E.A., Zhuvak O.V., Jokinen T., Heinisuo M., 2016
