Chapter 2 Literature review summary E q u a t i o n C h a p t e r ( N e x t ) S e c t i o n 1
-The resistance function of the Eurocode for M-V interaction EN 1993-1-1, CL. 6.2.8 can be rewritten to a derivation by Drucker (Drucker, 1956). Drucker defined the derivation as an upper bound approach, and the derivation assumed that the section could withstand a plastic hinge. Furthermore, the derivation is based on Tresca’s yield criterion. These principles do not correspond with Eurocode’s view in general, and particularly not with the definition of class 3 sections. Therefore, it was recommended to formulate an alternative resistance function for M-V interaction in general.
-Contrary to EN 1993-1-1, CL. 6.2.8, Leth provided an M-V interaction resistance function based on equilibrium conditions with an elastic plastic boundary (Leth, 1954). The elastic-plastic stress distributions are applicable to class 3 I-sections. Moreover, Leth used the Von Mises yield criterion instead of Tresca’s yield criterion. Therefore, Leth’s derivation provides a good alternative to EN 1993-1-1, CL. 6.2.8.
-The resistance of I sections to instability is determined by the overall slenderness of the section. Even though the linear interpolation method for the governing slenderness proposed in AM-1-1-2012-02 may lead to safe design values if the definition of class 3 is reevaluated, an improvement can be made by incorporating both λf and λw in a single resistance function. Similarly, the current approximation for the class boundaries in EC3 and the corresponding amendment AM-1-1-2012-03 could be improved by incorporating both λf and λw in a single class boundary.
Chapter 3 Proposed theoretical approximation E q u a t i o n C h a p t e r 3 S e c t i o n 3
-Many M-V interaction methods including EC3 are based on a cross-section subdivided into separate areas, submitted to absolute shear- or axial stress. The Von Mises equation indicates that an area submitted to σ-τ in interaction is up to 41% more efficient compared to an area subdivided into separate σ-τ regions. Consequently, an upper bound solution for M-V interaction can only be obtained with σ-τ in interaction. The upper bound solution for M-V interaction was approximated numerically, and can be used as verification for other M-V interaction design functions.
-An analytical solution for M-V interaction was derived from an optimization of Leth’s derivation (Leth, 1954). The analytical solution is based on the equilibrium conditions and is valid with elastic-plastic axial stress distributions. Therefore, the analytical solution is applicable to class 3 I-sections. Contrary to the Eurocode, the analytical solution corresponded to the approximate upper bound obtained with the iterative approximation. Therefore, the analytical solution provided a good alternative to the Eurocode.
-The analytical solution was extended to take account of the fillets. The results for HEA, HEAA, HEB, HEM, and IPE were compared with each other. The datasets indicated that HE sections are less affected by shear stress than IPE sections, since their moment capacities are less dependent on the web.
Moreover, the IPE datasets indicated that the influence of shear stresses on the moment capacity increased with height.
Chapter 4 Numerical modelEquation C h a p t e r 4 S e c t i o n 4
-Even though the numerical analysis was geometrically and materially non-linear, no effect of the order of loading on the ultimate bending moment capacity was perceived.
-The linear buckling analysis is a widely used method to obtain imperfections as input for a geometrically and materially non-linear analysis with imperfections included. However, due to the difference in linear and non-linear material behavior, the eigenmode of the linear buckling analysis can be incompatible with the failure mode of the corresponding non-linear analysis. It was recommended that the Eurocode should explicitly state imperfections obtained from a linear buckling analysis are inapplicable if the eigenvalue exceeds the elastic resistance of the geometrically and materially nonlinear model.
-The Eurocode currently does not state any requirements for the minimal distance between load and support in a 4-point bending test to comply to beam theory. The American code ACI 318 section 10.7 prescribes two times the section height as minimum length between load and support, and was confirmed to be correct.
-The selected parameters of the FEM-model accurately describe the buckling behavior of the experiments in (Lay, Adams, & Galambos, 1965). Moreover, good accordance has been obtained between the constitutive behavior of the FEM model and displacements derived from theory.
Chapter 5 Numerical tests E q u a t i o n C h a p t e r 5 S e c t i o n 5
-Currently, Abaqus CAE is not capable to terminate an analysis after failure occurs. The termination program that was written in python is an adequate work around to terminate an analysis right after the maximum capacity is reached. Terminating the analysis after failure helps avoid unnecessary FEM-computations. Reducing computational time and storage.
-The current boundaries for class 3-4 in EN 1993-1-1, CL. 5.5 and the corresponding amendment AM-1-1-2012-03 must ensure an elastic bending moment capacity can be reached. However, at the current class 3-4 border moment capacities higher and lower than the elastic bending moment capacity have been obtained. Therefore, the current classification method is an inaccurate method to relate slenderness ratios to the elastic bending moment capacity.
-Limit values for flexural buckling can only be described with a combination of λf and λw. Therefore, the current classification method is insufficient. Alternatively, the border of class 3-4 can be described with
50.17 2.374e+05
f3.66
wIf this equation is satisfied, the section is within class 3, and an elastic bending moment capacity is ensured. If the slenderness is too high and the equation is not satisfied, the section is within class 4.
-The formula for ε in EN 1993-1-1.5 to determine the slenderness for S355 and S460 leads to increasing safe design values for increasing steel grade.
-Both the linear interpolation method according to AM-1-1-2012-02, and EN 1993-1-1, CL. 5.5 overestimate the bending moment capacity of class 3 steel I-sections. Moreover, the method is inappropriate for local buckling in class 3 since it does not include the combined effect of λf and λw. Alternatively, the bending moment capacity for class 3 steel I sections can be determined with the reduction factor:
3 ,
class M f w pl
M M
-For increasing flange thickness the current definition for Vpl increasingly overestimates the shear capacity of I sections.
-EN 1993-1-5 and EN 1993-1-1 use Vpl,Aw and Vpl, Av respectively. These different definitions cause an inconsistency on the threshold of reducing the shear capacity due to shear buckling. Eurocode should adopt a uniform definition for the shear capacity throughout both sections. Even though Vpl,Aw resulted in a better approximation for the shear capacity compared to Vpl, Av, it was still inaccurate.
-The FEM-results confirmed that the shear buckling resistance in EN 1993-1-5 is solely dependent on λw, in contrary to flexural buckling. However, the resistance function in EN 1993-1-5 is convex, whereas the FEM-results suggest a linear relation. Therefore, EN 1993-1-5 is an inaccurate method to determine the shear capacity of I shaped steel beams with a non-rigid end post. A more accurate approximation for the reduction factor for can be described as:
1
55.5 0.0034
V w w
3 ,
class pl Aw V
V V
-The governing failure loads for M-V interaction in class 3 are equal to the failure load for flexural or shear buckling. Therefore, M-V interaction according to EN 1993-1-1, CL. 6.2.8 does not need to be considered for class 3 sections. A schematization of the final model for MV-interaction is illustrated in Figure 87, that summarizes the main conclusions.
Figure 87- final MV-interaction model with:
1: Mclass3 2: No MV-interaction 3:Vclass3
Chapter 6 Post processing resultsE q u a t i o n C h a p t e r 6 S e c t i o n 6
-The regression line coefficient b and the coefficient of variation Vδ for the resistance function Mclass3
are well within their boundaries for all subsets. Therefore, the resistance function Mclass3 is an accurate approximation of the bending moment resistance of class 3 steel I-sections.
-The regression line coefficient b and the coefficient of variation Vδ for the resistance function Vclass3
are well within their boundaries for all subsets. Therefore, the resistance function Vclass3 is an accurate approximation of the shear resistance of class 3 steel I-sections with a non-rigid end post and without transverse stiffeners.
-The partial safety factor γM0 for Vclass3 and Mclass3 can be taken as 1.0 for S235, S355, and S460.
-The total test population was evaluated with radii to determine the deviation of the numeric result to the combined model of Mclass3 and Vclass3. The corresponding partial safety factor γM0 was equal to 1.0.
Therefore, it was statistically confirmed no M-V interaction function is required.
7.1 R
ECOMMENDATIONS-The resistance functions obtained from the results are applicable to rolled steel class 3-I sections.
Since the slenderness ratios in the test population incorporate slenderness ratios equal to those of HEA and HEB sections, it was suspected the resistance functions are valid for these sections too. The IPE 300 was used as the unreduced section, however, additional sections including HEA, HEB and HEM sections could be tested to ensure the design rules are generally applicable.
-The experimental results from (Lay, Adams, & Galambos, 1965) were used to validate the numerical model. Even though these results incorporated instability, the sections were not equivalent to class 3 sections. An attempt was made for validation with slender sections with the experimental results of (Holtz & Kulak, 1975), however, due to insufficient documentation the comparison between results remained too uncertain. Furthermore, experimental research on sections with slender webs and flanges seemed unavailable. Therefore, experimental research on combined slenderness ratios could contribute the validity of the FEM-model.
-The bending moment resistance was dependent on a combination of λw and λf. However, the test population was based on the current definition of class 3 sections. The test population was insufficient to propose a relation for the class 2-3 boundary, as it was estimated to lie in the range that is currently defined as class 2. Therefore, additional FEM-simulations need to be conducted to determine the threshold of slenderness for which section can no longer resist a plastic bending moment. The FEM-model of this research is a suitable method to continue the research.
-Since the bending moment resistance of class 3 section was dependent on λw and λf, the bending moment resistance of class4 sections will be too. Therefore, further research into the bending
moment resistance of class 4 sections was advised. The current methods and FEM-model can be used to continue the research for class 4 sections.
-During the research the definition for the shear capacity Vpl appeared to increasingly overestimate the shear capacity for increasing flange thicknesses. It was suspected that the current method overestimates the contribution of the flanges to the total shear capacity. Even though the shear capacity based on Aw provided a better approximation, still not a single section could achieve a plastic shear resistance. Therefore, further research into the true shear capacity of steel I-section is recommended.
-The decrease of the limit values for shearbukcling appeared to be linearly proportional to an increasing web slenderness. The Eurocode, however, assumed this relation is convex. Therefore, additional research for additional configurations with rigid end posts and transverse stiffeners can determine the overall shear buckling behavior.