4 Numerical model Equation Chapter Se ction
4.2 Validation
4.2.3 Conclusions Validation
The FEM model was evaluated on maximum capacities and constitutive behavior, with respect to theoretical and experimental results. The results are presented in Table 9.
Table 9- Deviation of the FEM model from experimental or theoretical results
ΔM/Mpl [%] Δδ [%]
Experimental results 1,33 <5
Mechanics 0,35 0,01
Good accordance of the FEM models with theoretical and experimental results had been obtained.
However, further validation with experimental tests of non-compact sections in a 4-point bending is advised. Based on current results it was concluded that the FEM model could be used for further research.
0 0,2 0,4 0,6 0,8 1 1,2
0 1 2 3 4
M/Mel[-]
ϴ/ϴpl[-]
WS-12-N-ABAQUS WS-12-N-Experimental
5 N UMERICAL TESTS
EQUATION CHAPTER 5 SECTION 5In this chapter the numeric analyses are evaluated. First, the structure of the numeric model is described. The method that was used to modify the slenderness ratios of the section, and the method to determine the loading is explained. Furthermore, the concept and structure of the pre- and post-processing in Python is explained. Subsequently, the test population and the results are discussed.
5.1 G
ENERAL PROPERTIES NUMERIC MODELThe variables that were determined in the parametric are summarized in Table 11. A static general step was used to obtain node coordinates that are superimposed on the node coordinates of the GMNIA. The GMNIA is used to obtain the bending moment capacity for a slenderness and M-V-load combination. The leading parameters in this research were the slenderness ratios, they are defined according to (5.1) and (5.2) for web and flange respectively.
Table 10- Summary of FEM properties of the numerical tests
FEM Software Abaqus CAE 6.14-4
Data processing software Python 2.7
Buckling analysis Static general
GMNIA Riks analysis
Amplitude imperfection flange ϴf=1/15 Amplitude imperfection web e0w= hw/200
Element type C3D8I
Material model Bilinear true stress strain
Youngs modulus 2,1E5 N/mm2
Poisson’s ratio 0.3
Steel grade S235/S355/S460
Mesh section 4 elements over thicknesses
Maximum mesh aspect ratio 1:15
Symmetry axis No
5.2 G
LOBAL GEOMETRYUnbraced lengths in section a as illustrated in Figure 45, were used to induce local buckling. Lateral torsional buckling was suppressed by BC-LT, which is located in the middle over the full length of the flanges. A pinned (BC1) and roller (BC2) support were used to model a simply supported beam. The boundary conditions(BC) are listed in Table 11.
Rigidbody constraints were used to model the stiffeners above the reference points (RP) of the loads and supports. The rigidbody connection in combination with the BC acted as a ball joint. The moments on RP1 and RP2 were conveyed to the surface of the ends of the beam through rigidbody connections as well. An overview of the constraints is presented in Table 12.
The position where results were measured are summarized in Table 13. The bending moment was measured with the sectionprint command in S2 and S3. To ensure this command functioned correctly, the bending moment was derived from loading, and reaction forces as well.
Table 11- Overview of boundary conditions
Type Lateral restraint Rotational restraint
BC-LT Line X -
BC1 node XYZ -
BC2 node XY -
Table 12- overview of geometric constraints
Master Slave Constraint type
RP1 S1 Rigidbody
RP2 S4 Rigidbody
RP4 SET2 Rigidbody
RP5 SET3 Rigidbody
BC1 SET1 Rigidbody
BC2 SET4 Rigidbody
Table 13-Overview of locations of obtained results
Results Location
Momentmid S3
MomentLoad S2
Reaction forces BC1, BC2
Madd RP1, RP2
Loading RP4, RP5
DeflectionMID RP3
5.3 S
ECTION GEOMETRIESNo common section type comprises the dimensions that represent the full range of class 3, therefore, a basic section is selected and the slenderness was increased. The basic section was determined by geometry and loading. The bending moment induced by the concentrated loads was countered by the bending moments at the ends of the beam. However, to what extend could be countered was limited to Mpl. Therefore, if the concentrated loads induced a moment larger than 2·Mpl, the beam would fail prior in bending before obtaining Vpl. The basic section was selected for a geometry that could develop the highest shear force, without exceeding 2 times the design bending moment reduced for that shear force according to the Eurocode. This criterion is described by equation (5.3). The results in Table 14 indicate that the IPE 300 could obtain high V/Vpl ratios, and would be an average representation of the total IPE population. Therefore, the IPE 300 was selected for further analyses.
2,5 h V 2 M
y V Rd, , (5.3)Table 14- Maximum shear capacity for IPE Sections with a=2,5h
The thickness of the flanges was reduced from the outer tips inwards to keep the slenderness of the web constant, the thickness of the web was reduced symmetrically as illustrated in Figure 46. The fillet radius was determined by matching the obtained thickness of the flange and web to existing sections and their corresponding radius. A match was obtained through the average of a tf-radius and tw-radius pair, rounded to an existing radius.
0,780,8
5.4 L
OADING5.4.1 Static general loading
To obtain the imperfections points on the flange were displaced up to the required amplitude, as described in chapter 4.1.8.
5.4.2 GMNIA loading
All tests were loaded with two concentrated loads (P1, P2), and two moments (M1, M2) at the ends, therefore the model was force controlled. P1, P2 were applied as a traction force on the surface of the FC-DB-DI model as illustrated in Figure 25 to surface S2, S4. The bending moments were applied to RP1 and RP2 according to Figure 45. Even though no order effects were perceived in the parametric study both concentrated loads and moments were increased simultaneously. To obtain the bending moment capacity at a target shear force current design codes were used to estimate the required loading. As illustrated in Table 14 certain sections could not obtain Vpl due to a minimal length between load and support. Consequently, if the target shear force exceeded the estimated shear capacity, the target shear force was reduced. The principle of the combinations of the concentrated loads and the bending moment for increasing shear force is illustrated in Figure 47.
To transfer the bending moment from RP1 and RP2 to the beam a rigid body constraint was used. The rigid body is used to model a ball joint, and keeps the end surface of the beam in plane. Just coupling the rotations would be insufficient as solid elements do not have rotational degrees of freedom.
Pure bending moment, no shear force.
P1 and P2 cause a bending moment below failure. M1 and M2 are used to increase the bending moment up to failure.
P1 and P2 would had resulted in failure due to bending. M1 and M2 are used to counter premature failure in bending.
M1 and M2 have reached their maximum at the supports. No higher shear force can be obtained.
Figure 47-Load configuration for increasing shear force
5.5 T
EST POPULATIONThe test population was subdivided into multiple categories with respect to steel grade, slenderness, and M-V loading. The amount of FEM analyses in each test population is illustrated in Table 15, Table 16, Table 17. for S235, S355, S460 respectively.
S235 298
S355 114
S355 (without scaling) 42
S460 105
Total 559
To determine the slenderness ratios for the test population the proposed c/t ratios for classification in AM-1-1-2012-03 were considered. No restrictions are prescribed by the Eurocode for the slenderness of the flange if the web is classified as class 3 and vice versa. Therefore, the limits of slenderness were taken from the least slender IPE section, up to the threshold of class 4 sections. The slenderness ratios of an IPE 300 were added and subsequently the ranges were linearly subdivided.
The boundaries of the classes shift by ε with increasing yield stress. To evaluate the influence of the yield stress on the results the subdivision of classes (measuring points) was kept constant. Therefore, the geometries of higher steel grade sections were increased by ε to match the shifting class boundaries. To research the influence the yield stress a subset of S355 was added without scaling, as is presented in Table 18.
235 fy
(5.4)
The borders of the slenderness classes were examined more thoroughly. The effect of slenderness on the load carrying capacity was expected to be highest for class 3 slenderness ratios, therefore, these ratios were emphasized.
To research the influence of shear stress on the moment-capacity sections were tested with 6 load cases spread linearly from V/Vpl=0 to V/Vpl=1. Results for S235 indicated no decline up to V/Vpl=0.5, therefore, the intermediate measuring point V/Vpl=0.2 was omitted for S355 and S460. To further clarify the behavior of the model V/Vpl=0.75 and V/Vpl=0.9 were added in the S235 test population.
Even though no criteria were specified for the ratio λw/λf with respect to class 3, the flange must be able to withstand the shear stress at the flange web junction. Therefore, NEN-EN 1993-1-1 relates the surfaces of flange and web in chapter 6.2.6. with equation (5.5). Only if results for M-V interaction were divergent, and did not meet this criterion, they were removed from the test population.
f 0.6
w
A
A (5.5)
Table 15- Amount of FEM analyses in test population of S235
Table 16- Amount of FEM analyses in test population of S355
web Class1 Class2 Class3
Table 17-Amount of FEM analyses in test population of S460
web Class1 Class2 Class3
Table 18-Amount of FEM-analyses in test population of S355 without scaling
S355 web Class1 Class3 Class 4
Flange λ 20 35 53 72 83 92.5 102 111.5 121
Class1
4 0 0 0 0 0
5.28 1 0 1 0 1
7 1 0 1 0 1
Class2 8 1 0 1 0 1
Class3
9 1 0 1 0 1
10 0 1 1 1 1 1 1 1 1
11 0 0 0 0 1 0 1 0 1
Class4
12 0 1 1 1 1 1 1 1 1
13 0 0 0 0 1 0 1 0 1
14 0 1 1 1 1 1 1 1 1
5.6 A
PPROACH ON SCRIPTINGThe pre- and post-processing of the FEM analyses was automated using python 2.7. This chapter elaborates on the general set-up and explains where to find all specific information.
5.6.1 General script design
Batch files were used to run series of FEM-analyses without the need of intervention. The Batch file ran every model twice; once to call Abaqus cae and run the FEM-analysis, once to export the results.
This consecutively structure ensured results were extracted even when the FEM analysis was aborted prematurely, and would had terminated the python script. This structure is illustrated in Figure 48.
Figure 48- Concept of general script design
Multiple components of the script repeated once or several times. Repetitive sections were extracted from the main script to improve readability, and reduce lines. These modules were called in the main script through importing their definitions. The created modules are listed and their functions are explained:
-Runanalysis (main script called by Batch file) APPENDIX XVI.I
First, all required models were imported and the basic parameters were defined. Subsequently the loading was derived from the target MV-ratio. Based on the loading and geometry the name of the analysis was created and the corresponding folders were created. The script created identical models for the static general step and the GMNIA. The static general step was executed and the deformed node coordinates were stored in a node file. Subsequently, the coordinates in the node file were superimposed on the geometry of the GMNIA model. After the imperfections were applied, the GMNIA was executed. Simultaneously, a background program was invoked that regularly read the result file.
5 Increments after the programs detects the derivative of the σ-δ graph was negative, the program sent an abort command using the command prompt. The possibility to terminate the analysis after failure greatly reduced computation time, and avoids unnecessary data storage. After termination all data was retrieved from the ODB. Subsequently, the data, charts, and images of the failure modes were exported to an excel file.
-Geometry APPENDIX XVI.II
Determined all global and local geometries, translated slenderness to section dimensions and returned them.
Createbeam APPENDIX XVI.III
Contained all information on the FEM model, assembly, and instance. Createbeam was called twice in the main script to ensure the models of the static general step and GMNIA were identical.
-ReadODB APPENDIX XVI.IV
Contained all methods to extract data at nodes(sets) and elements(sets) from the ODB.
-ReadDAT APPENDIX XVI.V
Was used to loop over the .dat file to extract results.
-Strainlimit APPENDIX XVI.VI
Checked how many elements exceeded the plastic strain of 15%. If more than 7 elements exceeded the plastic strain limit the corresponding increment was returned, and flagged in the excel file.
-Createchart APPENDIX XVI.VII
Was used to generate excel charts from the retrieved data.
-Eigenmodeimage APPENDIX XVI.VIII
Contained the method to export images from the static general step and the GMNIA. Images were exported on the maxima of the σ-δ graph and the final increment.
5.7 R
ESULTSThe results of the FEM-analyses are displayed in this chapter and are compared to their
corresponding design codes in EC3. The design codes and corresponding amendments that were discussed in the problem statement are evaluated for S235, S355 and S460. The relevant design codes were:
EN 1993-1-1, CL. 5.5 Bending resistance reduced for local buckling EN 1993-1-1, CL. 6.2.8 M-V interaction, without influence of instability EN 1993-1-5, CL. 5.1 Shear resistance reduced for shear buckling
Since the class boundaries are closely related to the resistance functions for instability the class boundaries were evaluated as well. Furthermore, the implications of scaled geometries for higher steel grades is discussed. To improve clarity all the legend entries of the graphs in this chapter correspond to the results that are summarized in tables in the appendices.
5.7.1 Bending moment in class 3 sections S235
The results for the maximum bending moment capacity were plotted against the increasing slenderness for flange and web in Figure 49, and Figure 50 respectively. The graphs of the total test population were presented in APPENDIX XXIII. In Figure 49, the increasing flange slenderness showed a quadratic decline up to the border of class 3 (λf=10), and subsequently, appeared to continue to decline linearly.
Figure 49-Limit values of the bending moment capacity for increasing λf in S235
In Figure 50, the increasing web slenderness caused a steep decline for the bending moment capacity between λw=35 and λw=72. Subsequently, the steep decrease appeared to level out between λw=82 and λw=121. In contrary to the quadratic shape for λf in Figure 49, the shape of the graphs for λw in Figure 50 were more similar to a cubic function.
Figure 50-Limit values of the bending moment capacity for increasing λw in S235
The graphs of Figure 49 and Figure 50 both illustrate that the bending moment capacity was dependent on a combination of λw λf, and cannot be determined with a single slenderness parameter. If the bending moment capacity is determined with a single slenderness nonetheless, both Figure 49 and
0,84
Comparison with design rules with a single slenderness parameter λ in class 3
To compare the results for the bending moment capacity with EN 1993-1-1, CL. 5.5 and AM-1-1-2012-02 the bending moment capacity was rescaled with (M-Mel)/(Mpl-Mel). This scale is based on Mel=0 to Mpl=1. First, the results with a single slenderness parameter in the class 3 range were evaluated.
Both Figure 51 and Figure 52 indicate EN 1993-1-1, CL. 5.5 is a conservative method, except for (λw=72 λf=14). Moreover EN 1993-1-1, CL. 5.5 does not seem to describe the physical behavior of buckling, and confirms AM-1-1-2012-02 was correct to propose an alternative method. However, except from λf=4, AM-1-1-2012-02 does not lead to safe design capacities. Moreover, the linear interpolation from Mpl to Mel does not fit the curves of the results either. The results in both Figure 51 and Figure 52 illustrate both slenderness parameters influence the bending moment capacity, even though one of them is not class 3. Physically this is a logical result, i.e., even though a class 1 or 2 plate might not buckle itself, it restrains the plate that does buckle. Both slenderness parameters contribute to the overall buckling resistance of a member. Therefore, both AM-1-1-2012-02 and EN 1993-1-1, CL. 5.5 are unsuitable methods to describe the physical behavior of buckling properly.
Figure 51- Limit values of the bending moment capacity reduced for λf compared to design codes Class in 3
Figure 52-Limit values of the bending moment capacity reduced for λw compared to design codes Class in 3 -0,2
Comparison with design rules with both slenderness parameter λf and λw in class 3
The results for λf and λw both in class 3 and corresponding design codes are illustrated in Figure 56 and Figure 57. Both figures illustrate EN 1993-1-1, CL. 5.5 overall gives conservative results if either λf or λw
is in the class 3 range, similar to Figure 51, and Figure 52. However, if λf and λw are both in the class 3 range the elastic bending moment capacity is not achieved and EN 1993-1-1, CL. 5.5 overestimates the bending moment capacity. As was stated in the problem statement, Eurocode 3 defines Class 3 sections as:
‘class 3 cross-section are cross-sections in which the elastic moment can be reached, yet local buckling prevents the development of a plastic moment’
According to this definition, the borders of class 3 for λw and λf are currently overestimated. In AM-1-1-2012-03 it was already proposed to reduce the border of λw, class4 from 124, to 121. However, the results in Figure 57 indicate λw=121 does not ensure the elastic bending moment is obtained either.
Therefore, the proposal in AM-1-1-2012-03 cannot be regarded as a safe limit value. To ensure an elastic bending moment is obtained for class 3 sections, either the boundaries for class 3 must be reduced to λf=9 and λw=65, as is illustrated in Figure 56 and Figure 57. Or alternatively, the class 3 border must be described by a combination of λf λw.
To illustrate this effect the results were 3D plotted on a scale of (M-Mel)/(Mpl -Mel), for λw and λf. Subsequently, the graph is cut of at Mel, representing the border of class 3 to 4. In Figure 55 the cutting plane of Mel is visible in the upper right corner. The actual border of class 3-4 seems to be a convex function for λw and λf. Points on these cutting planes can be subsequently curve fitted to define the boundaries of class 3.
The border of class 2-3, i.e., the slenderness for which a plastic moment can be obtained seems to be defined by a similar convex function. However, it passes through the range that is currently defined as class 2 which was not in the test population. Therefore, too little data was available to confirm this assumption.
From Figure 55 it could be concluded that the class 3 border was best described by both λw and λf. The use of both slenderness parameters would lead to a more economic design compared to straightforwardly using λf=9 and λw=65.
Figure 53-Limit values for the bending moment with respect to λf λw with class 3 borders and Mel 3D view
Figure 54-Limit values for the bending moment with respect to λf λw S235 top view
Figure 55-Limit values for the bending moment with respect to λf λw in S235 cut of at Mel top view
Furthermore, the linear interpolation method proposed in AM-1-1-2012-02 uses the governing slenderness λ, therefore, it does not incorporate the composite effect of λf or λw on the bending moment capacity either. Consequently, if both λf or λw are in the class 3 range the interpolation method overestimates the bending moment capacity even worse compared to EN 1993-1-1, CL. 5.5. From these results it was concluded both EN 1993-1-1, CL. 5.5 and AM-1-1-2012-02 do not lead to safe limit values.
Therefore, an alternative design rule for local buckling of class 3 steel I sections that incorporates both λf and λw was required.
Figure 56-Bending moment capacity reduced for λf with λw in class 3 compared to design codes
Figure 57-Bending moment capacity reduced for λw with λf in class 3 compared to current design codes -1
5.7.2 Bending moment capacity in class 3 for S355 without scaled geometry and results
The influence of the steel grade on the bending moment capacity was tested with and without scaling the flange and web by ε. The results for unscaled sections in S355, were compared to their geometrically identical sections in S235 in Figure 58 and Figure 59. Moreover, additional graphs for local buckling are added in APPENDIX XXIII. The slenderness of the sections with an increased steel grade was a factor ε higher than their S235 equivalent, therefore, it was expected the normalized
The influence of the steel grade on the bending moment capacity was tested with and without scaling the flange and web by ε. The results for unscaled sections in S355, were compared to their geometrically identical sections in S235 in Figure 58 and Figure 59. Moreover, additional graphs for local buckling are added in APPENDIX XXIII. The slenderness of the sections with an increased steel grade was a factor ε higher than their S235 equivalent, therefore, it was expected the normalized