M-V interaction S235 Normalized to buckling loads

The graphs for M-V interaction in Figure 65 and Figure 64 were rescaled to their measured failure loads in Figure 72 and Figure 73. The tables and graphs of the total test population are presented in APPENDIX XIX, APPENDIX XX respectively. Every section in the M-V population could not withstand a plastic bending moment. Therefore, it was assumed failure occurred in elasto-(plastic) buckling, and the failure loads were defined as Mbuck and Vbuck. Normalizing the results to their failure load Mbuck and Vbuck illustrates the relation between both types of instability. The definitions for Mbuck and Vbuck could be taken as the design values according to EN 1993-1-1, CL. 5.5 and EN 1993-1-5, CL. 5.1 respectively.

However, both methods had been disputed previously. Moreover, the use of design loads instead of true failure loads will blur the relation of the physical behavior of M-V interaction. Therefore, Mbuck

and Vbuck were taken as the failure loads obtained from the analyses.

Similar to Figure 65 and Figure 64 the graphs in Figure 72 and Figure 73 show either local or shear buckling is governing, and the transition is sudden. Therefore, M-V interaction seems to be determined by Mbuck and Vbuck, without interaction. In chapter 1.1, a round curve in Figure 3 was suggested for M-V interaction in class 3, the results indicate the curves should had been squares.

Figure 72-Limit values M-V interaction normalized to corresponding buckling loads S235 0

6 P OST PROCESSING RESULTS

EQUATION CHAP TER 6 SECTION 6

This chapter translates the conclusions from the results to proposals for design rules. Subsequently the design rules are statistically evaluated according to SAFEBRICTILE (Taras, Dehan, Simões da Silva, Marques, & Tankova, 2014)

6.1

PROPOSED DESIGN RULES

According to the previous conclusions in chapter 5.7, proposals were made for:

-The bending moment capacity of class 3 steel I beams -The shear capacity of class 3 steel I beams

-M-V interaction for class 3 steel I beams - The class 3-4 boundary

6.1.1 Bending moment capacity for class 3 steel I sections

In chapter 5 it was concluded the bending moment capacity for class 3 steel I sections should be a function of λf and λw, as is illustrated by equation (6.1). To find a function for the reduction factor χM

the limit values as a factor of Mpl were curve fitted in Matlab R2016a as a function of λf and λw. A polynomial function for χM was evaluated for increasing powers on the R-square value, the results are presented in Table 19.

 

3 ,

class f w M pl

M    M (6.1)

Table 19, Evaluation of polynomial curve fit based on R-square value

Curve fit Power of λf Power of λw R-square

1 1 1 0.8772

2 1 2 0.9707

3 1 3 0.9867

4 2 1 0.9534

5 2 2 0.9823

6 2 3 0.9948

7 3 1 0.9550

8 3 2 0.9955

9 3 3 0.9955

As was concluded in chapter 5.7.1, the influence of λf on the bending moment capacity appeared quadratic and the influence of λw appeared cubic. Therefore, it was expected a polynomial function with λf2 and λw3 resulted in the highest R-square value. Even though higher degree polynomials resulted in higher R-square values, Curve fit 6 showed the most logical fit. The function for Curve fit 6 is described by equation (6.2), and is illustrated in Figure 74.

2 2 2 2 3

1 2 3 4 5 6 7 8 9

( , )

_{f w M}

C C

_f

C

_w

C

_f

C

_{f w}

C

_w

C

_{f w}

C

_{f w}

C

_w

                      

(6.2) With

C1 = 0.9828 C4 = -1.949e-05 C7 = -4.668e-06 C2 = 0.005523 C5 = -0.0002801 C8 = 1.719e-06 C3 = 0.001553 C6 = -1.25e-05 C9 = 1.51e-08

Figure 74-Polynomial curve fit with λw3 and λf2 including the class 3 boundaries in red

The function is valid within the range results were obtained. Therefore, the sections must be classified as class 3 according to EN 1993-1-1, CL. 5.5. Moreover, the slenderness of the flange must comply to

6.1.2 Shear capacity for Class 3 steel I sections

Based on the limit values for shear buckling a linear approximation can be derived, as described by (6.3). The linear approximation can be used for beams with a non-rigid end post and without transverse stiffeners. Moreover, the slenderness needs to comply to 55.5>λw>121. The linear approximation provided a better fit for the limit values compared to EN-1993-1-5.5.1, as is illustrated in Figure 75.

Consequently, the linear approximation is safer for lower λw, and more economic for higher λw.

  

¹



55.5 0.0034

 

V w w

 

 



  (6.3)

3 ,

class pl Aw V

V V  (6.4)

Figure 75- Limit values class 3 shear capacity with design codes 0

0,2 0,4 0,6 0,8 1 1,2

72 82 92 102 112

V/Vpl,Aw[-]

λ_w [-]

[λf·ε=10]-S235 [λf·ε=12]-S235 [λf·ε=14]-S235 EN 1993-1-5 Linear approximation

6.1.3 Evaluation Design codes M-V interaction

The approximations for Vclass3 and Mclass3 were evaluated for S235, S355, and S460. The axes were scaled to M/Mclass3 and V/Vclass3. Therefore, every value equal to 1 means the approximation is accurate. Moreover, every value above 1 indicates the approximation leads to safe design values.

The limit values in Figure 76 and Figure 77 indicate Mclass3 is accurate in S235, and leads to

increasingly safe design values for S355 in Figure 78, Figure 79, and S460 in Figure 80, and Figure 81.

The tables and graphs of the total test population are presented in APPENDIX XXI, and APPENDIX XXII respectively. This conclusion was in line with chapter 5.7.4, where it was already concluded the value for ε was a little conservative. Furthermore, a similar trend was observed for Vclass3. The flange slenderness was not incorporated in Vclass3, therefore, a small spread was observed in the results for deviations in λf. However, the maximum deviation to V/Vclass3=1 in Figure 76 is around 3.5% on the safe side, which was not considered significant, and a problem. Furthermore, the graphs illustrate the total test population can safely be approximated with either Mclass3 or Vclass3, therefore, it was proposed to exclude M-V interaction resistance functions for class 3 steel I sections.

Figure 76-M-V interaction limit values normalized to Vclass3 and Mclass3 S235

Figure 77-M-V interaction limit values normalized to Vclass3 and Mclass3 S235 0

Figure 78-M-V interaction limit values normalized to Vclass3 and Mclass3 S355

Figure 79-M-V interaction limit values normalized to Vclass3 and Mclass3 S355

Figure 80-M-V interaction limit values normalized to V and M S460 0

Figure 81-M-V interaction limit values normalized to Vclass3 and Mclass3 S460 0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 M/Mclass3

V/V_class3

[λf·ε=10, λw·ε=72]-S460 [λf·ε=10, λw·ε=83]-S460 [λf·ε=10, λw·ε=102]-S460 [λf·ε=10, λw·ε=121]-S460

6.1.4 Redefining the boundaries of class 3

As was proposed in chapter 5.7.1, the boundaries of class 3 are currently insufficient and should be defined through both λf and λw. This chapter evaluates the results and aims to redefine the boundaries of class 3 by relating them to both slenderness parameters.

Boundary of class 3-4

The boundary of class 3 to 4 is defined by sections that can obtain an elastic bending moment resistance. To obtain results at Mel, values for λw and λf were linearly interpolated. Subsequently, MATLAB 2016a was used to curve fit the slenderness ratios. The boundary of class 3 was best described by equation (6.5) with an R-square value of 0.9923. The proposed class3-4 boundary is illustrated in Figure 82

50.17   2.374e+05  

_f^3.66

 

_w (6.5)

Figure 82-Results for Mel and Mpl and the proposal for redefining class 3-4 boundary

Boundary of class 2-3

The boundary of class 2-3 is defined by sections that can only just withstand a plastic bending moment.

However, the only results that withstood Mpl were obtained at λf =4 and λw=20. The actual boundary probably lies between (λf =4, λw=92.5) and ((λf =11, λw=20) as illustrated in Figure 82. Therefore, the boundary lies in the range that is currently defined as class 2. No analyses were conducted in the class 2 range; therefore, too little results were obtained to properly describe the boundary of class 2-3.

0 20 40 60 80 100 120

0 2 4 6 8 10 12 14

λw[-]

λ_f [-]

Results for Mpl Results for Mel Class 2-3 boundary Class 3-4 boundary

6.2 S

TATISTICAL ANALYSES

This chapter evaluates Mclass3 and Vclass3 statistically according to Safebrictile (Taras, Dehan, Simões da Silva, Marques, & Tankova, 2014) to obtain a partial safety factor γM0. This partial safety factor takes account for model uncertainties and dimensional deviations. To prove no significant M-V interaction was apparent the total test population should be described with either Mclass3 or Vclass3. Based on the graphs in chapter 6.1.3 results were divided into the test population of either Mclass3 or Vclass3.

6.2.1 Statistical procedure

According to safebrictile, the following steps were followed:

Step1:

The basic input parameters were determined. Therefore, the slenderness parameters λf ·ε and λw ·ε were broken down to the following variables:

X1=b

Subsequently, the theoretical strength determined with Vclass3 and Mclass3 was compared to the experimental strength obtained from FEM-results. For preliminary design the regression line coefficient b should not diverge strongly from unity and the coefficient of variation Vδ should not exceed 10%.

b Regression line coefficient

re FEM limit value

rt Resistance function limit value

n Number of experiments in test population

exp( ) 12

Step 3:

This step evaluates the sensitivity of the resistance function to the basic input variables, by computing the error propagation term Vr, t2.

   

Vr,t2 error propagation term

Xi Basic input variable

rm,t =gr,t(Xm) resistance function evaluated with the mean values of the basic input variables

k Total number of basic input variables

σj standard deviation of basic input variables

The mean values and coefficients of variation of the basic input variables are presented in Table 20, and were obtained from (Simoes da Silva, et al., 2017).

Table 20-Statistic properties basic input variables

fy,nom [N/mm²] mean/nom [-] c.o.v. [-]

The log-normal variation coefficients were determined according to:



²



The design value of the resistance rd is dependent on the amount of FEM results n in the test population, and calculated according to (6.12), (6.13).

For n<100

With design fractile factors k obtained from Table 21.

Table 21-Values for k according to EN 1900-Annex D

n 1 2 3 4 5 6 8 10 20 30 ∞

Vx,known 4.36 3.77 3.56 3.44 3.37 3.33 3.27 3.23 3.16 3.13 3.04

Subsequently, the partial safety factor γM* was determined with (6.14).

, ,

The obtained safety factor over the target safety factor should be below the acceptance limit fa.

*

The experimental results were compared to theoretic results with respect to Vclass3 and Mclass3

separately, and in terms or radii. The principle is illustrated in Figure 83. If radii are used, Mclass3 and Vclass3 are regarded as one model to describe M-V interaction. However, if M-V interaction is excluded beforehand, the experimental results can be compared to either Mclass3 or Vclass3 directly.

0

6.2.2 Statistical analysis Mclass3

The results of the statistical analysis for Mclass3 were obtained through the method described in chapter 6.2.1. The results for Mclass3 are presented in Table 22, the graphs of the regression lines are presented in APPENDIX XXIV. The design rule Mclass3 was evaluated in subsets for pure bending moment(V=0) and for the total test population (Shear stress included) in steel grades S235, S355 and S460. All subsets easily complied to 0.85<b<1.15 and Vδ <10%, therefore, the resistance function Mclass3 can be considered a good approximation of the limit values. Furthermore, the partial safety factor γM* is below the acceptance limit of γM0=1.0 for every subset as is illustrated in Figure 84, therefore, γM0=1.0 can be used for the total test population of Mclass3.

Table 22-Results statistical evaluation Mclass3

n b Vδ Vr,t2 Vr γM* fa(γm0=1.0)

6.2.3 Statistical analysis Vclass3

Analogue to the statistical evaluation of Mclass3 the test population of Vclass3 was subdivided into steel grades S235, S355, and S460. The results for Vclass3 are presented in Table 22, the graphs of the regression lines are presented in APPENDIX XXIV. Each individual subset complied to 0.85<b<1.15 and Vδ <10%, therefore, the resistance function Vclass3 can be considered a good approximation of the limit values. Furthermore, the partial safety factor γM* is below the acceptance limit of γM0=1.0 for every subset as is illustrated in Figure 85, therefore, γM0=1.0 can be used for the total test population of Vclass3.

Table 23-Results statistical evaluation Vclass3

n b Vδ Vr,t2 Vr γM* fa(γm0=1.0)

S235 87 0.993 0.027 0.00418 0.0707 0.970 1.053

S355 34 1.022 0.020 0.00377 0.0646 1.019 1.048

S460 35 1.042 0.031 0.00310 0.0633 1.000 1.048

6.2.4 Statistical analysis Mclass3 Vclass3 with radii

Finally, the total test population was subdivided into steel grades S235, S355, and S460. The results for the statistic evaluation with radii are presented in Table 22, the graphs of the regression lines are presented in APPENDIX XXIV. Each individual subset complied to 0.85<b<1.15 and Vδ <10%, therefore, the resistance functions can be considered a good approximation of all limit values and confirms no M-V interaction needs to be considered. Furthermore, the partial safety factor γM* is below the acceptance limit of γM0=1.0 for every subset as is illustrated in Figure 86, therefore, γM0=1.0 can be used for the total test population.

Table 24-Results statistical evaluation of Vclass3 and Mclass3 in radii

n b Vδ Vr,t2 Vr γM* fa(γm0=1.0)

S235 298 0.999 0.0088 0.00407 0.0644 0.972 1.048

S355 114 1.018 0.013 0.00225 0.0419 1.000 1.037

S460 105 1.032 0.017 0.00153 0.0427 0.968 1.032

Figure 84- Acceptance limit graph of the results with respect to Mclass3

Figure 85-Acceptence limit graph of the results with respect to Vclass3

0,94

7 C ONCLUSIONS

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  

_f^3.66

 

_w

If 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.

-The resistance functions obtained from the results are applicable to rolled steel class 3-I sections.

In document Eindhoven University of Technology MASTER Shear-moment interaction on class 3 rolled I-shaped steel sections Overdorp, B. (pagina 67-0)