Evaluating Structural Engineering Finite Element Analysis Data. Using Multiway Analysis

Transcription

1 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC Evaluating Structural Engineering Finite Element Analysis Data Using Multiway Analysis Matija Radovic Civil Engineering Department University of Delaware Newark, DE. Jennifer McConnell Civil Engineering Department University of Delaware Newark, DE. Abstract- The scope of this paper is to introduce multiway analysis into structural engineering research and to outline methodology used in tensor decomposition of finite element analysis (FEA) data. More specifically, the example evaluated herein evaluates the stress distribution of two different highway bridge structural components (girders and cross frames), being subjected to incrementally increasing forces. Additionally, the paper shows potential advantages of using multiway methods in interpretation of FEA data and makes recommendations for future investigations on the use of multiway methods in FEA postprocessing of structural engineering data. Keywords Multiway analysis, finite element analysis, bridge engineering, structural engineering, Tucker3 model. I. INTRODUCTION In current structural engineering practice, finite element analysis (FEA) is commonly used to predict the structural behavior of various structural members, of assemblies of structural members, and / or of entire structures. For example, FEA can be used to determine the maximum strength or displacements of a structural member under various loading scenarios or to investigate the distribution of stress between various members. It is routinely used to design unique structures such as bridges and buildings and as a research tool. FEA is based on a discretization of structural parts into geometric shapes (elements) that are bounded at their corners or edges with points (nodes). Each element has assigned material properties. The grid lines seen in the left of Fig. 1 denote the element boundaries for an I-shaped member. The response of these elements to an input loading is calculated via a system of partial differential equations with each element containing multiple unknown quantities that are calculated by the FEA method [1]. Since the solution for the set of the partial differential equations is based on numerical approximations, a more detailed set (a finer FEA mesh with a larger number of elements) will theoretically yield more accurate solutions up to a point where the influence of mesh size converges to a common solution. The number of elements in a typical model could vary anywhere from hundreds to millions. As a result of a typical FEA, displacements, stresses, and strains in multiple directions are computed for each element. Furthermore, one FEA may Fig. 1. Finite element model (FEM) of a steel bridge I-girder (on the left) shows the element mesh, where rectangles represent discretized geometrical shapes (elements) that can be numerically modeled with a system of partial differential equations. Stress contours (on the right) the spatial variation of stress magnitudes due to imposed loads. Red colored elements are elements with the largest stresses, while the green colored elements are the elements with the lowest stresses. realistically contain anywhere from one to hundreds of loading conditions, producing a unique data set for each loading. Thus, the potential output from these analyses is immense. Fig 2. shows a subset of potential FEA data, showing one type of output (von Mises stresses) for one part (bottom flange) of one highway bridge member (one girder) that was subjected to 6 load increments. In current practice, only a small fraction of this available data is quantitatively analyzed. For example, it is often the case that only the extreme values in the data set (such as minimum and maximum stresses or maximum displacements) at a particular region of interest are analyzed. A possible exception to this statement are contour plots showing the spatial distribution of the magnitudes of a specific output variable that can be produced by some FEA post-processing software (as seen in Fig 1. on the right). However, an approach for performing a rigorous quantitative analysis of the data has yet to be applied in any published works that could be identified by the authors. For these reasons, this paper explores the use of big data applications to analyze a large data set resulting from the FEA of a representative civil engineering structure, a steel bridge. The general goal of this effort is to first determine a methodology Center for Innovative Bridge Engineering (CIBrE), University of Delaware, Newark, DE.

2 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC using big data analysis techniques to improve upon the current practice of using isolated subsets of the available data and/ or visual data contours that lack a means to readily conduct Stresses (psi) Load increments Fig. 2. FEA output of bottom flange element stresses from bridge girder, subjected to 6 load increments. Each line represents a single bottom flange element and each point on the line represents the stress value measured at increasing load increments. quantitative data analysis. Secondly, it is desired to assess the practical significance of this data, it being anticipated that a more rigorous quantitative analysis will have the potential to reveal a greater understanding of the structural response, for complex structures in particular. Another benefit in more comprehensive analysis of the large FEA data sets is envisioned as being a means to compare and contrast similarities and differences in competing design options. Therefore the scope of this paper is to explore the use of multiway data analysis techniques in analyzing structural engineering FEA output. More specifically, the example evaluated herein assesses the stress distribution of two different structural components (girders and cross frames) in highway bridges that are subjected to incremental loading. Tucker3 tensor decomposition is used as an analytics tool to explore the dataset resulting from the FEA of such a steel I-girder highway bridge. This paper outlines the methodology used in Tucker3 tensor decomposition to propose a new methodology for interpreting FEA data in structural engineering, compares the outputs of traditional interpretation of FEA analysis to interpretation of FEA analysis based on multiway models, and makes recommendation for future use of multiway tools in bridge engineering practice. II. BACKGROUND Generally, big data is a term used to describe large and complex data sets. The main benefit in using large data sets is that larger data sets may lead to more accurate and insightful understanding of a particular phenomenon. Typically, engineers and analysts usually impose a 2-dimensional structure on data analysis which has diminished capability to reveal valuable patterns and trends in the data. Due to the sheer volume and complexity of a large data set, sophisticated analytical tools are necessary to fully utilize it. Multiway data analysis is one such tool that has become popular as an exploratory analysis tool in discovering the latent structures in higher-order datasets [2,3]. Higher-order data sets are data sets with more than two modes (i.e., dimensions), with a mode representing a set of data arranged in matrix form. Conventional data analysis deals with analyzing data in the form of two dimensional matrixes. The addition of a third mode will result in the data being in the form of a data cuboid, which is a typical example of a higher-order data set. For example, in the present situation of interest, consider a data set that consists of stress values along the length of a structural member. If we assume that the member consists of a top flange (TF), a bottom flange (BF), and a web, then mode 1 in the data matrix will be the elements comprising these member components in each cross-section of the member (with mode 1 usually organized as the row variables) and mode 2 will be the distances along the length of the girder (usually organized as the column variable). The data inside the matrix would consist of stresses at each cross-section element at each distance. If behavior under increasing magnitudes of load is of interest, then a third mode could consist of load magnitudes (usually the third mode is called a tube in 3-D array). In models with increasing complexity, additional modes can be added. For example, the influence of temperature fluctuations on the structural member s stresses, during incremental loading, could serve as a 4th mode in the present example. Depending of the complexity of the problem as many modes as needed can be added. Clearly, multiway or higher-order data is ubiquitous and there are several methods for analyzing such data. One of the methods available for the analysis of multiway data is higher order singular value decomposition (HOSVD), also known as Tucker tensor decompositions [4]. If the data set is in a threeway format, Tucker3 tensor decomposition is a special case of HOSVD used for analyzing the data. Tucker3 tensor decomposition is mathematically expressed in matrix (1) and element form (2), which show this to be a trilinear decomposition method, decomposes the three-dimensional array into sets of scores (or loadings) that potentially describe the data in a more condensed form than the original data array. Here X is a three-way data set; G is the core tensor array whose entries show the level of interaction between the different loading matrices; A, B, and C are loading matrixes and E is an error array. Consequently, in element form, g is the core tensor array entry; a, b, and c are the loading matrix element entries; e is an entry from the error array; and P, Q, and R are the numbers of components in the loading matrices A, B, and C, respectively. Tucker3 decomposition of the structural engineering example (1) (2)

3 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC discussed in the previous paragraph is conceptually presented in Fig. 3. In order to properly decompose three-way array using Tucker3 method, several core tensors (arrays) must be fitted in order to find the most appropriate one, given the dataset under In order to properly decompose three-wayarray using Tucker3 method, several core tensors (arrays) must be fitted in order to [G]pxqxr find the most appropriate one, given the dataset under analysi s. analysis. Fig. 3. The Graphic biggest representation advantage of three-way of using tensor Tucker3 decomposition tensor using Tucker3 method. decomposition method is its ability to compress variation, extract features, explore data, and generate parsimonious models, especially from highly correlated data sets [5]. The biggest disadvantage of using Tucker3 tensor decomposition method is that it sometimes requires complex data interpretation before optimal model parameters are chosen. Parameters in questions are the size of the core tensor G (p x q x r); core tensor and loading matrix constraints, such as orthogonality or non-negativity; and the selection of an optimization algorithm based on the data structure and / or software used. Due to these restrictions, the user must be familiar with the practical significance of the data set of interest in order to interpret and conduct the analysis in meaningful and accurate manner. III. METHODOLOGY The investigation and results presented in this paper are based on the FEA of a representative civil engineering structure, a steel I-girder highway bridge. This structure, the description of the corresponding FEA model, and the loading conditions applied to the model are first described below in Sections A through C, respectively. Section D then describes the post-processing of the FEA data that was needed as preprocessing for executing the tensor decomposition. This methodology section concludes with explaining the use of the Tucker3 method in tensor decomposition. A. FEA Subject Bridge Description The FEA results which will be presented in the following sections of this paper were based on a representative highway bridge, labeled Bridge 7R. A photograph of this structure can be seen at the top of Fig. 4. This bridge is selected due the extensive prior evaluation of the behavior of this structure, including in-service and destructive testing and corresponding FEA [6]. The bridge is a three-span, 63 skewed steel I-girder bridge, with the span of interest being 15ft. long. The bridge has 4 plate girders spaced 8 ft. on center. All four plate girders were fabricated from A7 steel having a web depth of 6 in. and a web thickness of 3/8 in. The top flange dimensions were 2 x 1 in. for the exterior girders and 18 x 7/8 in. for the interior girders. The effective dimensions of the bottom flanges at mid-span, including the cover plates that existed in this location, were 2 x 3-1/8 in. for the exterior girders and 2 x 2-1/2 in. for the interior girders. The web was stiffened with double-sided transverse stiffeners on the interior girders, in addition to single-sided transverse stiffeners and single-sided longitudinal stiffeners on the exterior girders. The composite 8 in.-thick concrete deck and 2 in. haunch were connected to the girder with shear connectors. Lateral bracing consisted of K-frames that were spaced at 2 ft. along the length of the girders, with the first cross-frame being offset 22.5 ft. from the supports. These were composed of 4 x 3 ½ x 3/8 angles and connected to gusset plates, with four bolts at the end of each member, which were in turn bolted to transverse stiffeners welded to the girders. Additionally, end diaphragms were used to connect the girders at each end of the member. These diaphragms were steel I-sections of equal depth to the girders. B. FEA Description Previous work [6] has validated a FEA of Bridge 7R. In this FEA, all bridge components were modeled using a relatively fine mesh of reduced-integration shell elements. Longitudinal and transverse deck reinforcement was modeled using a builtin function in the finite element software. Full composite action between the reinforced concrete deck and the steel girder was developed by using a beam-type multipoint constraint that assures nodal compatibility between the girder top flange and the deck at these locations. The cross-frame-togirder connection is modeled by merging nodes on the crossframes and transverse stiffeners serving as connection plates. Merged nodes are also used to connect the connection plates, webs, and flanges. Linear elastic properties are used to model the concrete deck while for the steel, a multi-linear constitutive response is used, simulating the linear elastic, yield plateau, strain hardening, and ultimate regimes, based on the von Mises yield criterion [7]. The bottom section of Fig. 4 shows a cross-sectional view of the finite element model (FEM) used in the FEA of Bridge7R.

4 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC C. FEA Loading Conditions The structural response of Bridge 7R was examined using the modified Riks method. The basic function of this method is that it applies progressively larger magnitudes of load until an ultimate capacity is reached while searching for a combination of forces and displacements (and thus other C. FEA Loading Conditions The basic function of the modified Riks method is that it Fig. 4. Bridge 7R: photograph of actual structure on the top and crosssectional view of FEA on the bottom. associated structural response metrics such as stress) that satisfy equilibrium at each magnitude of loading. The Riks method is ideally suited to analyzing the behavior after such a peak loading is reached. Such algorithms exist in commercial FEA software, such as Abaqus 6.12 [7], which is used in this work. In this case, the load is expressed in terms of load proportionality factors (a multiple of the input load). For this study, the input load is a HS-2 vehicle load commonly used in bridge design and evaluation [8]. Since each LPF is a multiple of the input load and the input load is a HS-2 loading vehicle, the results of the Riks method are expressed in terms of number of these design vehicles. For the scenario being described here, for which non-linear geometry was prescribed, the maximum LPF at which convergence is obtained is 17. D. Data Preprocessing for Tensor Decomposition In order to carry out Tucker3 decomposition, the data must be organized in a specific three-way format. In this work, the goal was to investigate the stresses in different element groups representing different structural members under increasing loads. Since the girders of Bridge 7R were simply supported composite beams, their bottom flanges will experience the highest stress. Thus, only the bottom flange element stresses were considered for the girders in this example. It follows that girder bottom flange and cross-frame element groups, stress magnitude (expressed in terms of the number of elements experiencing the same range of stress at a given load) and increasing loads were selected to represent the dimensions in the array. Stress ranges were used because it would be impractical to capture all possible stress values of each element in the structure. In this way, the data emulates the form of stress histograms for each loading increment. The data extracted from the FEA was organized into element groups representing different structural members of the bridge in different locations in the bridge structure. For example, cross-frame elements that are connecting Girder 1 and Girder 2 (see Fig. 4) are labeled XG1; cross-frame elements that are connecting Girder 2 and Girder 3 are labeled XG2; and cross-frame elements that are connecting Girder 3 and Girder 4 are labeled XG3. Girder 1 bottom flange elements are labeled G1BF, Girder 2 bottom flange elements are labeled G2BF, and so on. At the end of this step, a total of 7 element groups are formed (X1G, X2G, X3G, G1BF, G2BF, G3BF and G4BF), each one containing histogram data for 51 stress ranges and 17 load magnitudes, representing data from 28,976 locations in the structure. The optimum histogram bin size was determined using the mean integrated squared error procedure [9]. Mean integrated square error algorithm is used to measure the goodness-of-thefit of a histogram by computing the estimated errors for several candidate bin sizes, and the bin size that yields the smallest estimated error is selected. The stress ranges (histogram bins) were kept constant for all element groups as well as being constant between load increments. As a result, for the data set presented in this paper, the optimal bin size was determined to be 733 psi and a total of 51 bins were used. Note that von Mises stresses [7] are used in this work, so all stress values have a positive sign. Because the girder bottom flanges and cross-frames have significantly different numbers of elements, comparing counts of the frequencies of elements in each stress bin would not reveal the desired information. Therefore, the number of elements in each stress range in each element group is normalized (divided) by the number of elements in the corresponding group. For example, if 665 is the number of bottom flange elements in Girder 4 that are in a given stress range and because the total number of elements in the bottom flange of Girder 4 is 3838, then the resulting value in this stress range (bin) would be.173 (665/3838). The procedure is repeated for each element group and for each loading. This preprocessing results in creating element group stress histograms as shown in Fig. 5. All preprocessing was executed by a custom-made Matlab code [1].

5 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC E. Tensor Decomposition Procedure using Tucker3 Method Tensor decomposition of the resulting three-way array, mathematically expressed as xϵr ( ), was performed using the N-way Toolbox in Matlab R214. The first mode of the array is made up of 7 element groups (4 girder element groups and 3 cross-frame element groups organized in rows), the second mode consists of 51 stress ranges (51 stress histogram bins organized in columns), and the third mode consists of the 17 loading increments also known as LPFs. Fig. 5. Stress profiles of seven element groups. Longitudinal axes consist o increments (LPFs) and vertical axis represents frequency of occurrence exp the corresponding element group. G1 BF G1 BF.15 % of total elements % of total elements G2 BF psi LPF LPF LPF LPF.3.25 G3 BF G4 BF LPF LPF LPF LPF XG XG XG3.6 1 LPF LPF LPF LPF LPF LPF

6 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC (72.91%) Fig. 6. Percentage of variance explained by Tucker3 decomposition of different core array sizes. Horizontal axis represents model dimensionality, p x q x r. TABLE I. LOADING SCORES OF CORE ARRAYS FOR TUCKER3 (2, 2, 2) MODELS IN UNFOLDED FORM C1 C1 C2 C2 B1 B2 B1 B2 A A Since there were no constant terms (offsets) in the data sets that needed to be removed, no data centering was performed during analysis. Using Tucker3 decomposition, data set x is decomposed into three loading matrices (A, B, C) and a core array (G) as shown in (1). Orthogonality constraints were imposed on the loading matrices. The core array was unconstrained (meaning there were no limits imposed on how the values in the core array were calculated), while the dimensions of the core array were selected based on two criteria, percentage variance explained and optimal model complexity. Percentage variance explained shows how well the decomposed model fits the original data, where models with higher percentages of variance explained indicate more accurate models. The optimal model complexity is the one that requires the smallest number of components but still captures a high percent of the variance in the model [12]. A total of 27 core tensor arrays with different complexities and resulting in different percentages of variance explained were computed and analyzed. Fig. 6. graphically shows the results of the analysis. Models with large variance explained but with larger core array size were ultimately excluded because they were less interpretable. For example models with core arrays size of p x q x r > 12, had high values of explained variances, but these models also have larger number of components that needs to be interpreted. It is also noted that higher variance captured can be the result of the overfitting (making the model unreasonably complex, such as having too many components relative to the number of observations) and not necessarily an ideal model for a given data set. Therefore the 2x2x2 model was selected because it captured 72.91% of variance with the least degree of complexity (core array size is 8). Comparing this result to the variance captured in prior work from other fields, this value is deemed reasonable, falling within the range of values previously reported. Table I shows the core array loading scores for the Tucker (2, 2, 2) decomposition model. These loading scores are given in unfolded form, which is simply rearranging the three-way array into matrix format. The significance of these values is discussed in the following section. IV. RESULTS The loading scores (matrix entries) for Factor A (element groups) are displayed in Fig 7. These results show that that cross-frame groups XG2 and XG3 have high negative loading scores in Component 1, while girders groups have high positive loadings scores in Component 2. The high positive scores in Component 2 for Factor A correspond to the widest spread in stress distribution, while the high negative scores in Component 1 correspond to the narrowest spread in stress distribution. The stress profiles show that bottom flange of Girder 4 (see Fig 5, plot G4 BF), has the widest spread of stress distribution of all groups; the tensor decomposition confirms this visual conclusion by quantifying that the bottom flange of girder (G4 in Fig. 7) also has the highest positive score for Component 2 of all element groups. Conversely, cross-frame element group XG2 (Fig. 7) has the highest negative scores in Component 1 and narrowest spread of stress distribution as it can be seen in Fig. 5. This shows one means in which multiway analysis can be used to readily quantifying differences in the behavior of or load sharing between different members. The loading scores for Factor C (LPFs) are presented in Figure 8. Results show that high positive scores in Component 1 and high negative scores in Component 2 correspond to low LPFs, while high positive score in Component 2 correspond to large LPFs. To further explore this behavior, a hierarchical clustering of the factor C loadings was performed. The results are presented in the form of a dendogram in Fig. 9. A dendogram is a tree like diagram used to exemplify group clustering. The horizontal arrangement of the branches in the dendogram indicates which points are most similar to each other. The height of points in the dendogram indicates the degree of difference among the points. The dendogram shows that the Factor C scores are clustered in two main groups, with one primary branch (when viewed from the top down and marked by red dashed rectangle in Fig. 9) consists of LPFs 1 to 4 and the second primary branch consists of LPFs 5 to 17. This suggests a deviation in behavior between LPF 4 and LPF 5 (which is also marked by the dashed rectangle in Fig. 8). The height of these two branches of the dendogram also indicates a large difference between the two clustering groups. These results were somewhat unexpected because a linear elastic response of all element types considered is expected in this range of loading. Furthermore, the visible stress

7 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC distributions shown in Fig. 5 fail to display any anomalies throughout the loading range. In order to further investigate this finding, the results of the tensor analysis were compared to field data from Bridge 7R [13]. The bottom flange results from Girder 2 and Girder 3 are shown in Fig. 1, which indicate that after LPF 4 these load versus strain curves demonstrate an abrupt change in slope. This suggests that the present multiway analysis approach may provide a starting point for understanding this experimental phenomenon that has yet to be explained in prior work. The current state of the art of FEA post-processing involving visual inspection and evaluation of selected quantities lacks the ability to provide any significant explanation for these field results. V. CONCLUSION This paper has presented a new and innovative method of FEA data interpretation in structural engineering. More Fig. 7. Plot of loading scores of Factor A (element groups) Fig. 1. Bottom flanges field test results of girder 2 (G2 BF) and girder 3 (G3 BF) show significant deviation in girder response after load increment that corresponds to LPF 4 Fig. 8. Plot of loading scores of Factor C (LPFs) LPFs Fig. 9. Dendogram of factor C loadings. Red rectangle shows the cluster of first 4 LPFs specifically, the Tucker3 tensor decomposition method was used to explore stress distribution assessment of two different structural components (girders and cross frames) resulting from the FEA of a steel I-girder highway bridge. A Tucker (2, 2, 2) decomposition model was able to quantify and differentiate stress profiles belonging to different bridge components. This finding signifies the advantage that tensor decomposition methods have over traditional approach in interpretation of FEA results in bridge engineering, especially in terms of comparing stress distribution among different bridge components. Application of this method could have practical significance. For example recent studies [14, 15] investigated the effects of cross-frame designs on bridge behavior by analyzing FEA data of steel I-girder bridges using traditional FEA data interpretation approaches. In both studies, variations in stress distributions of different structural members were of interest. The ability to readily quantify and differentiate the stress profiles of different bridge components would be useful tool for researchers when assessing different design options. Furthermore, with the tensor decomposition methods presented in this study, researchers could be able to quantify and statistically differentiate behaviors of different bridges. Future work in assessing the effectiveness of tensor decomposition methods in comparing stress distributions between different bridges could also have significant practical benefit for providing information to optimize future highway bridge designs.

8 214 IEEE International Conference on BigData (IEEE BigData 214), Washington DC Additionally, the Tucker tensor decomposition method shows promise for exposing the latent behavior of bridge components subjected to increasing load. More specifically, a Tucker (2,2,2) model identified a difference in bridge response that was also found in field testing results, while traditional FEA post-processing (analysis of maximum stresses in element groups) failed to do so. Whether this is coincidence or pointing to an underlying change in structural response that is possible to detect via multiway analysis requires further evaluation to determine. REFERENCES [1] Bathe, K.-J. (1982). Finite element procedures in engineering analysis. Englewood Cliffs, N.J: Prentice-Hall. [2] Comon, P. (29). Tensor decompositions, state of the art and applications. ArXiv Preprint arxiv : [3] Acar, E., Bro, R., & Schmidt, B. (28). New exploratory clustering tool. Journal of Chemometrics, 22(1), [4] Kolda, T. G., & Bader, B. W. (29). Tensor decompositions and applications. SIAM Review, 51(3), [5] Bro, R. (1998). Multi-Way Analysis in the Food Industry: Models, Algorithms, and Applications, Doctoral Dissertation, Københavns Universitet Københavns, Universitet. [6] McConnell, J., Chajes, M., & Michaud, K. (214). Field testing of a decommissioned skewed steel I Girder bridge: Analysis of system effects. ASCE Journal of Structural Engineering, in press. [7] ABAQUS 6.12 User Documentation. (214). Simulia. [8] American Association of State Highway and Transportation Officials. (212). AASHTO LRFD Bridge Design Specifications, 5th Edition. American Association of State Highway and Transportation Officials, Washington, D.C. [9] Shimazaki, H., & Shinomoto, S. (27). A method for selecting the bin size of a time histogram. Neural Computation, 19(6), [1] Van, V. L. H. (197). Materials science for engineers. Reading, Mass: Addison-Wesley Pub. Co. [11] Matlab R214. (214) Mathworks. [12] Singh, K. P., Malik, A., Singh, V. K., & Sinha, S. (26). Multi-way data analysis of soils irrigated with wastewater A case study. Chemometrics and Intelligent Laboratory Systems, 83(1), [13] Michaud, K (211). Evaluating Reserve Bridge Capacity Through Destructive Testing of a Decommissioned Bridge, Master Thesis, University of Delaware. [14] Radovic, M., McConnell, J. (214). Evaluation of Cross -frame Designs for Highly Skewed Steel I-Girder Bridges. Proceedings of 31 st Annual International Bridge Conference. IBC 14-47, [15] McConnell, J., Radovic, M., and Ambrose, K. (214). Cross-Frame Forces in Skewed Steel I-Girder Bridges: Field Measurements and Finite Element Analysis. University of Delaware Center for Bridge Engineering, Final Report to Delaware Department of Transportation.