Eindhoven University of Technology
MASTER
Shear-moment interaction on class 3 rolled I-shaped steel sections
Overdorp, B.
Award date:
2018
Link to publication
Disclaimer
This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required
Eindhoven University of Technology
REPORT
Shear-moment interaction on class 3 rolled I-shaped steel sections
Members of graduation committee:
prof.ir. H.H.Snijder ir. R.W.A.Dekker
dr.ir. P.Teeuwen (Witteveen+Bos)
Author:
B. Overdorp 0776853 June, 2018.
Final thesis in partial fulfillment of the requirements for the master program Architecture, Building and Planning.
Specialization Structural Design
This document is the main report of the graduation project. This report is accompanied by a literature review and appendices, and combined constitute the total research.
A-2018.229 O-2019.229
S UMMARY
This research is conducted in response to a proposal for amendments to the Eurocode (Greiner, et al., 2009). The amendment AM-1-1-2012-02 proposes a linearly decreasing bending resistance for steel I- sections in class 3, as defined in EN 1993-1-1, CL. 5.5. Furthermore, in AM-1-1-2012-03 a discrepancy between the slenderness limits for webs in EN 1993-1-1 and plates in EN 1993-1-5 is indicated.
However, the amendments did not specify how moment-shear (M-V) interaction should be approached. Therefore, it was assumed that the current design code for M-V interaction EN 1993-1-1, CL. 6.2.8 should be applied. Closer inspection of the current M-V interaction design rule suggested the resistance function assumed that stresses could be redistributed according to plasticity, i.e., it suggested the resistance function was not suitable for class 3 I section.
Therefore, this research focused on determining the influence of M-V interaction on class 3 rolled steel I-sections. Resistance functions for the bending moment capacity, shear capacity, and intermediate M- V interaction were determined. The safety of the design rules was evaluated, through computing the partial safety factor M0.
A literature study was conducted to provide basic insight into the background of classic M-V interaction theories. Furthermore, it was discovered that the Eurocode’s M-V interaction design rule EN 1993-1- 1, CL. 6.2.8 could be rewritten to a derivation of Drucker (Drucker, 1956). The derivation indicated that EN 1993-1-1, CL. 6.2.8 assumed plastic material behavior, and Drucker defined the derivation as an upper bound approach. Therefore, it was confirmed that EN 1993-1-1, CL. 6.2.8 was an unsuitable method for class 3 sections, and for M-V interaction in general.
Subsequently, an alternative view on stress distributions using Von Mises equation was used to approach an upper bound solution. An iterative optimization algorithm was used to determine the most efficient stress distributions, and resulted in an approximate upper bound solution.
Subsequently, a new analytical solution for M-V interaction was derived based on a derivation by Leth (Leth, 1954), that was applicable to class 3 sections.
The research continued with a numeric study to determine M-V interaction with instability. A 4-point bending test was modelled with the finite element method. More recent research (Shokoushian & Shi, 2014) suggested the resistance of class 3 I-sections is dependent on the combined slenderness of flange and web. The numeric results for the bending moment capacity confirmed this conclusion.
Therefore, an alternative resistance function for the bending moment capacity of class 3 steel I- sections as a function of the slenderness of the web and flange was proposed. The results from the numeric simulations indicated that the elastic bending moment capacity was not obtained for sections with both slenderness ratios in the class 3 range, therefore, both EN 1993-1-1, CL. 5.5 and AM-1-1- 2012-02 showed to be unsafe. An alternative relationship for the class 3-4 border including both slenderness ratios was proposed.
Furthermore, the results for M-V interaction indicated that the definition for Vpl increasingly overestimated the shear capacity for increasing flange thickness. Therefore, an alternative for the shear capacity of class 3 steel I-sections was proposed. The numerical limit values for M-V interaction indicated that either the shear capacity or bending moment capacity was governing for class 3 sections.
Consequently, no M-V interaction appeared to influence the resistance of the sections. To confirm this assumption the total test population was statistically evaluated with the proposed resistance functions for the shear capacity and bending moment capacity. The corresponding partial safety factor γM0 was equal to 1.0, therefore, M-V interaction does not need to be considered for class 3 steel I-sections.
C ONTENTS
Summary ... 3
Nomenclature ... 7
1 IntroductionEquation Chapter 1 Section 1 ... 10
1.1 Problem statement ... 11
1.2 Aim... 13
1.3 methods ... 13
2 Literature review summary Equation Chapter (Next ) Section 1 ... 14
3 Proposed theoretical approximation Equation Chapter 3 Se ction 3 ... 16
3.1 Iterative approximation ... 17
3.2 theoretical approximation ... 20
3.2.1 Analytical approximation ... 20
3.2.2 Numerical approximation ... 24
4 Numerical modelEquation Chapter 4 Se ction 4 ... 25
4.1 Parametric study ... 25
4.1.1 Geometry and setup ... 25
4.1.2 Elements ... 26
4.1.3 Symmetry axis ... 27
4.1.4 Mesh size section ... 28
4.1.5 Mesh size aspect ratio ... 29
4.1.6 Force and displacement controlled Loading ... 30
4.1.7 Load Application ... 31
4.1.8 Imperfections ... 34
4.1.9 Material model ... 39
4.1.10 Conclusions parametric study ... 40
4.2 Validation ... 41
4.2.1 with theoretic values ... 41
4.2.2 Validation with experiments ... 42
4.2.3 Conclusions Validation ... 44
5 Numerical tests Equation Chapter 5 Se ction 5 ... 45
5.1 General properties numeric model ... 45
5.2 Global geometry ... 46
5.3 Section geometries ... 47
5.4.2 GMNIA loading ... 48
5.5 Test population... 49
5.6 Approach on scripting ... 51
5.6.1 General script design ... 51
5.7 Results ... 53
5.7.1 Bending moment in class 3 sections S235 ... 54
5.7.2 Bending moment capacity in class 3 for S355 without scaled geometry and results ... 59
5.7.3 Bending moment capacity in class3 for S355 without scaled geometry with scaled results ... 60
5.7.4 Bending moment capacity in class 3 sections for S355 S460 with scaled geometry and results ... 61
5.7.5 M-V interaction S235 ... 62
5.7.6 M-V interaction S235 S355 S460 ... 64
5.7.7 M-V interaction S235 Normalized to buckling loads ... 66
6 Post processing resultsEquation Chapter 6 Se ction 6 ... 67
6.1 proposed design rules ... 67
6.1.1 Bending moment capacity for class 3 steel I sections ... 68
6.1.2 Shear capacity for Class 3 steel I sections ... 69
6.1.3 Evaluation Design codes M-V interaction ... 70
6.1.4 Redefining the boundaries of class 3 ... 73
6.2 Statistical analyses ... 74
6.2.1 Statistical procedure ... 74
6.2.2 Statistical analysis Mclass3 ... 77
6.2.3 Statistical analysis Vclass3 ... 77
6.2.4 Statistical analysis Mclass3 Vclass3 with radii ... 77
7 Conclusions ... 79
7.1 Recommendations... 83
8 Bibliography ... 84
APPENDIX I Iterative solver upperbound solution 88 APPENDIX II Derivation M-V-curve with elastic plastic boundary 89 APPENDIX III Derivation analytic theoretic approximation V/Vpl<2/3 92 APPENDIX IV Derivation analytic theoretic approximation V/Vpl>2/3 93
APPENDIX V Stress distribution and M-V interaction 94
APPENDIX VI Example individual M-V interaction curve 99
APPENDIX VII M-V interaction curves with fillets included 104
APPENDIX IX Results parametric study 111
APPENDIX X Derivation displacement and curvature 134
APPENDIX XI Validation with theoretic values 136
APPENDIX XII Lay adam and galambos tables and figures 137
APPENDIX XIII Derivation measurement location 140
APPENDIX XIV Experimental validation 143
APPENDIX XV Numeric model material model 147
APPENDIX XVI Script 148
APPENDIX XVII Numeric results normalized to plastic capacity - tables 166 APPENDIX XVIII Numeric results normalized to plastic capacity - graphs 170 APPENDIX XIX Numeric results normalized to buckling loads - tables 173 APPENDIX XX Numeric results normalized to buckling loads - graphs 176 APPENDIX XXI Numeric results normalized toMclass3 andVclass3- tables 182 APPENDIX XXII Numeric results normalized toMclass3 andVclass3- graphs 185 APPENDIX XXIII Numeric results flexural buckling - graphs 191 APPENDIX XXIV Statistic evaluation - regression line graphs 195
N OMENCLATURE
A Area of cross-section
Av Shear area
Aw Web area
D
Height of the plastic core in pure shearE
Young’s modulusG Shear modulus
I
Second moment of inertiaL
Beam lengthM
Bending moment0
,
plM M
Plastic bending resistance,
M
pl x Plastic bending resistance of elementN Normal force
P
Point loadQx log-normal variation coefficients
S First moment of inertia
U Strain energy
Ud Distortional energy density
Uh Dilatational energy density
U0 Strain energy density
V Shear force
0
,
plV V
Plastic shear resistance,
Vr t Error propagation term
V Coefficient of variation
W Section modulus
b Section width
b Regression line coefficient
e0w Amplitude geometrical imperfection of the web
f
y Yield stressfa Acceptance limit
h Section height
hw Clear web height
k Total number of basic input variables
kd design fractal factors k
n Amount of experiments
tx Thickness of element
r Radius of fillet
rd The design value of the resistance
re Experimental limit value
,
rm t resistance function evaluated with the mean values of the basic input variables
rt Resistance function limit value
s standard deviation of Δ
z1 Plastic-elastic boundary
Shear strain
M Partial safety factor
Deflection Logarithm of the error term
w Slenderness web
f Slenderness flange
Axial straineng Engineering strain
true True strain Rotation
Reduction factor due to shear stressj Standard deviation of basic input variables
x
, ,
y z
Normal stresses
VM Von Mises stressTr Tresca stress
0, , ,k Y Y
Uniaxial yield stress
true True stress
nom Engineering stressxy
, ,
xz yz
Shear stresses0 Yield stress in pure shear
Poisson’s ratio Reduction factor
1 I NTRODUCTION
EQUATION CHAPTER 1 SECTIO N 1In recent years, sustainability has become a significant topic for the building industry, forcing structural engineers to reduce material waste as much as possible. General wastage consists for 35%
of construction wastage, and from all cargo transport 25% is related to the building industry (Lichtenberg, 2012). Reducing the amount of required material benefits numerous environmental aspects of the building industry.
Material waste in steel structures has been effectively addressed with the optimization of steel sections. Optimization of industrialized products is the most effective method to ensure significant material economization, since they are frequently applied in the building industry.
To reduce material waste, structural designers are increasingly depending on higher steel grades. In current design code, steel grades above S235 may lead to a shift in class of the profile. If the section is moved to class 3, the design bending moment capacity changes from a plastic moment resistance to an elastic moment resistance, drastically decreasing its capacity. The boundary of class 2 and 3, theoretically cannot exist as it is now prescribed by the Eurocode, and is more likely to show a smoother transition. This transition of the plastic capacity to an elastic capacity in class 3 is now becoming an interesting field of research for material economy, as class 3 section are applied more frequently due to higher steel grades.
Shear moment interaction design rules in the Eurocode seem to be based on the presumption that the section is able to withstand a plastic moment, therefore, the present design code might not be applicable to class 3 sections.
Unless otherwise specified, the global coordinate system is defined as illustrated in Figure 1.
Figure 1-Coordinate system
1.1 P
ROBLEM STATEMENTThis research is conducted in response to a proposal for amendments to the Eurocode (Greiner, et al., 2009). This document proposes a linearly decreasing bending resistance for steel I-sections in class 3, according to Eurocode 3. Two amendments are relevant for this research, AM-1-1-2012-02 and AM-1- 1-2012-03, and will be referred to as first and second amendment respectively.
The first amendment (AM-1-1-2012-02) describes an alternative method for determining the moment resistance based on a c/t ratio in class 3 members. Present formulae only allow plastic material behavior in class 1 and 2, whereas class 3 and 4 rely on elastic material behavior. The proposal of this amendment is to linearly interpolate the moment resistance of a class 3 member, between class 2 and 4, with the c/t ratio as leading parameter. The moment resistance is subsequently reduced to take account of normal forces and bi-axial bending. This new transition allows for elastic-plastic material behavior in class 3, therefore, increasing the bending moment capacity. This new method is illustrated in Figure 2. Although the amendment only mentions MN-interaction, it may be assumed that shear stress reduces the bending moment capacity as is illustrated in Figure 2, using a similar method as is used to intercorporate additional normal forces, according to the design rules prescribed by the Eurocode.
Figure 2- moment resistance to c/t ratio
The second amendment (AM-1-1-2012-03) indicates the discrepancies between the reduction method for the strength of steel web and flanges, due to local buckling, and the analogue method for plates.
Moreover, the amendment indicates that present c/t values defining class 2, and 3, do not match FEM results, and are not safe sided. Combining both factors, this amendment aims to remove any discrepancies and adjust the c/t value that defines the boundary of class 3 and 4. Figure 2 is already based upon the proposed c/t value of the class 3 to 4 boundary.
In addition to local buckling induced by axial stress, class 3 members are prone to shear buckling as well. Moreover, the instability types are closely related through the geometry of a member, and cannot be suppressed separately. These types of instability are checked separately in current design codes corresponding to number 1 and 3 in Figure 3. However, the design code for M-V interaction (2) that should connect number 1 and 3, does not take account of instability. Therefore, number 2 applies to the upper circle. However, to describe M-V interaction in class 3 number 2 should be within the dark grey area.
1 EN 1993-1-1, CL. 5.5 Bending resistance reduced for local buckling 2 EN 1993-1-1, CL. 6.2.8 M-V interaction, without influence of instability 3 EN 1993-1-5, CL. 5.1 Shear resistance reduced for shear buckling
Figure 3- Current methods in Eurocode for M-V interaction and according instability
The Eurocode states that a shear force must be taken into account if the load reaches half of the plastic design capacity (Ved/Vpl,rd=0,5). Considering the yield criterion theory of Von Mises (Mises, 1913),the Eurocode must overestimate the moment capacity for small shear forces, and therefore, cannot be a lower bound solution. When shear forces exceed half of the plastic shear force capacity, Eurocode 3 prescribes two methods to calculate the interaction with a bending moment. The first method prescribes a reduced yield stress over the area affected by shear. The yield stress is reduced by a factor ρ, shown in (1.1).
2
,
2 Ed 1
pl Rd
V
V (1.1)
The structural designer must compute separate section moduli depending on whether the area is affected by shear. This method is very cumbersome to apply, which means it does not serve its purpose very well.
The second method distributes the stresses over separate areas of the cross-section, reducing the
2 ,
, ,
0
4
pl y w y
w y V Rd
m
W A f
M t
(1.2)
This method is only valid if forces may be distributed over the area. Class 3 members are defined by Eurocode 3 as follows: ‘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’. This implies that (1.2) is not applicable for class 3 and 4. Moreover, (1.2) reduces an area that has fully reached its yield stress, which is an overestimation if in that area stresses are elastic. Because the Eurocode gives no class requirement for any method, the resistance of class 3 sections may be calculated with methods based on unsuitable presumptions. If (1.1) shows to be inapplicable to class 3 sections as well, the Eurocode is also incomplete.
1.2 A
IMThis research focusses on determining the relationship of M-V interaction on class 3 steel I-sections.
Therefore, resistance functions for the bending moment capacity, shear capacity, and intermediate M- V interaction need to be determined. The safety of the design rules is evaluated, through computing the partial safety factor M0.
1.3
METHODSA literature study was conducted to provide basic insight into the background of classic M-V interaction theories. Furthermore, the derivation of Eurocode’s M-V interaction design rule (1.1) is obtained and evaluated. Subsequently, an upper bound approach and new resistance function for M-V interaction were proposed. These functions were used to compare with FEM-results.
M-V interaction will be described in threefold. The first design rule reduces the bending moment capacity for instability in class 3, similarly as proposed in AM-1-1-2012-02. Second, the shear capacity in class 3 needs to be evaluated. Finally, the M-V interaction between the previous 2 resistance functions needs to be determined. The safety factors M0 for the proposed design rules are determined through numerical data and a statistical analysis.
Numerical analyses were used to obtain the bending moment capacity of class 3 steel I sections that are submitted to M-V interaction. Prior to testing, the numerical models were validated using theoretical and experimental results. All numerical analyses were conducted using Abaqus CAE.
Research on instability requires a geometrically nonlinear analysis, therefore, methods for obtaining imperfections were evaluated. Subsequently, the RIKS method was used for a geometrically and materially nonlinear analysis with imperfections included (GMNIA). Python scripts were used as input for the Model Database (mdb), as well as to obtain output from the Output Database (odb).
The 4-point bending test contains a section that is loaded in pure bending, therefore, the effect of shear stress on axial stresses can be isolated. In prior research (Dekker, 2018) FEM-models have exceeded the theoretical plastic moment capacity. The 4-point bending test might suppress this phenomenon, and therefore, is selected as the basic setup. Two types of loading are applied to induce shear and axial stresses in the beam: two symmetrically placed concentrated loads, and two bending moments at both ends of the beam. Combinations of both loads are used to cover the entire range of the load resisting capacity. One roller and one pinned support are placed symmetrically at a distance
2 L ITERATURE REVIEW SUMMARY
EQUATION CHAP TER (NEXT ) SECTION 1In the literature review (Overdorp, 2018) the basic M-V interaction rules were covered, and the mechanics from which they originate were studied. The obtained overview aimed to relate the different approaches to the properties that define class 3 sections in the Eurocode. Compatibility between M-V interaction approaches and class 3 sections was found in derivations that are based on elastic and elastic-plastic stress distributions.
The literature review starts with the basis of M-V interaction, i.e., the underlying material behavior of Von Mises’s and Tresca’s yield criteria. Even though the material behavior on M-V interaction still must be determined empirically due to material impurity (Yu, 2004), Von Mises seems to provide a good approximation. The equilibrium conditions are another requirement that always need to be satisfied in M-V interaction. Therefore, equilibrium conditions were derived subsequently.
In literature, most researchers adopted the same approach. Stress distributions are derived from equilibrium conditions to ensure that stresses are statically admissible when the distribution is elastic or elastic-plastic. Subsequently, stresses are combined with a yield criterion to determine the maximum capacity in interaction. This approach was adopted by e.g. (Drucker, 1956), (Leth, 1954), (Horne, 1951) and (Neal, 1961). Alternatively, a plastic hinge is assumed and stresses are redistributed according to the view of the researcher e.g. (Onat & Shield, 1953), (Green, 1954) and (Hodge, 1956).
Local and non-local approaches on M-V interaction, upper and lower bound solutions have been obtained. From theory, it is undetermined whether local criteria provide a lower or upper bound solution. However, in comparison to other lower bound solutions local criteria seem to underestimate the load carrying capacity of a cross-section, and therefore, local approaches are not preferred By rewriting a derivation in (Drucker, 1956) the origin of the current method for M-V interaction in EC 3 was found. The method is derived for a rectangular cross-section, and assumes the moment capacity is not affected by shear stresses below V V/ 0 1 / 2 . However, the derived yield criteria indicate this cannot be true, and Drucker defined the method as an upper bound. The question is raised why the Eurocode would adopt a method that is disputed even by the author himself. The derivation is based on Tresca’s yield criterion, which is not in alignment with the view of the Eurocode. Moreover, Drucker assumed a plastic hinge and redistributed stresses. These stress distributions do not satisfy equilibrium conditions if stresses are partly elastic, and therefore, may only be used for determining limit loads in full plasticity. In Eurocode’s view of class 3 sections, plastic hinges cannot be obtained, and stresses may not be redistributed. Therefore, Drucker’s method is especially not applicable for Class 3 sections subjected to M-V interaction, justifying the need for an alternative method.
Additionally to Drucker’s first equation that was adopted by the Eurocode, Drucker derived an approach solely based on the equilibrium conditions and Tresca’s yield criterion. This method implies that the derivation is statically admissible and below the limit load. However, similarly to the first derivation flanges were not taken into consideration. Therefore, the method is applicable to I-sections up to the point where shear stresses reach the boundaries of the rectangleV V/ pl 2 / 3. According to Drucker, the exact solution should lie between his upper- and lower bound solution, corresponding to the non-local and local criterion respectively, as is illustrated in Figure 4.
Leth provided an interesting approach using the elastic plastic boundary, and this resulted in a good, lower bound solution which was valid up to . The Stress distributions are in agreement
criteria that constitute class 3 sections, more suitable than the present M-V interaction rules in EC3.
However, it might underestimate the load bearing capacity since the Von Mises stress is not equal to 1 over the full height of the cross-section. Therefore, Leth’s approach might be too conservative for class 1 and 2 sections. This assumption is supported if Leth is compared to the non-local criterion of Drucker, as is illustrated in Figure 4. To determine an M-V interaction formula for class 3 sections it was proposed to elaborate on Leth’s derivation. The aim is to derive a formula that agrees with equilibrium conditions, does not violate Von Mises, and is based on axial elastic-plastic stress distributions. All these criteria are currently not incorporated in the current EC3 M-V interaction rules for class 3.
In recent years, FEM studies have been conducted on the classification of I-shaped sections. Research disputes the use of separate slenderness criteria for buckling modes that are in fact related. Therefore, interaction of the slenderness ratios for web and flange is advised. However, absolute values for these slenderness ratios are based on contradictory results, and no conclusive answer can be obtained. Even though the influence of shear stress in these studies is often disregarded, shear buckling is closely related to the slenderness of the web and flange as well. Moreover, interaction between buckling modes induced by shear or axial compression is not excluded. Therefore, slenderness ratios for web and flange should be tested for load cases that include, and exclude shear stress. Additionally, in FEM analyses, fillets are commonly approximated with an increased shell thickness. However, this approximation leads to overlapping geometries, and solid elements should provide a better alternative.
3 P ROPOSED THEORETICAL APPROXIMATION
EQUATION CHAPTE R 3 SECTION 3This chapter continues on the literature review by evaluating the current method of the Eurocode, and elaborating on Leth’s derivation. In the method of the Eurocode, an optimization could be made with other stress distributions. The optimization is approximated with an iterative algorithm. First, the optimization is explained with N-V interaction, subsequently, it is extended to M-V interaction. The iterative approximation is an approximate upper bound solution to M-V interaction, and is used to verify the analytical solution. Subsequently, an analytical approximation based on Leth’s derivation is derived, and is evaluated numerically.
3.1 I
TERATIVE APPROXIMATIONTo compute the resistance of a section to shear and axial forces Eurocode’s method is based on the principle of subdivided areas that are separately loaded in shear- or axial stress, as illustrated by the blue rectangle in Figure 5. From this perspective, if an area is unloaded from shear stress, that same amount of area would become available to load in axial stress. Therefore, a decrease in shear stress is linearly proportional to an increase in axial stress. This approach is illustrated by the blue line in the graph of Figure 5. Alternatively, stresses can be applied in interaction according to the Von Mises yield criterion, as illustrated by the black rectangle in Figure 5. Since Von Mises yield criterion describes the maximal σ-τ interaction, Von Mises yield criterion itself is the upper bound solution for N-V interaction.
As discussed in the literature review, Von Mises equation is concave instead of linear. An equation for the largest square bounded by Von Mises graph in Figure 5 leads to an σ/τ interaction ratio that results in the most efficient use of area.
Subsequently, a comparison could be made between an area that loaded in shear and axial stress separately (blue line) or uniformly loaded with the optimal σ/τ interaction ratio (black line), as is illustrated for the web in Figure 5. When the surfaces of the blue and black rectangles were compared, or the shear and normal forces were computed, it shows that N-V in interaction is a factor 2 more efficient, equal to 41%.
int
1 2
eraction 2 vM
int
1 6
eraction 6 vM
1
seperated 2 vM
1 3
seperated 6 vM
1 1
2 4 2
seperated w seperated w vM
N h t ht
int int
2 1
eration w eraction 2 w vM
N h t ht
1 1 1
2 3 2 3
seperated w seperated w vM
V h t
ht
int int
2 1
eraction w eraction 2 3 w vM
V h t
ht
Figure 5- Comparison between a web loaded with σ/τ separated or σ/τ in interaction 0
0,1 0,2 0,3 0,4 0,5
0 0,2 0,4 0,6 0,8 1
τ/σvM[-]
σ/σvM [-]
However, this distribution only provides the highest axial force. To obtain the highest bending moment the influence of the lever arm must be considered. Since the axial stress is multiplied with the lever arm, the σ/τ ratio should shift towards σ away from the neutral line of gravity. An analytical solution for the most optimal distribution (upper bound solution) of shear and axial stress is complicated to derive, however, a script could loop over all stress distribution and select the highest moment iteratively. This script is added in APPENDIX I. To simulate this effect an arbitrary web area was divided in 8 elements from the neutral line of gravity outwards, these elements could be loaded in 6 σ/τ ratios that were selected on Von Mises’s graph as is illustrated in Figure 6. More elements and loading ratios would eventually converge to the true upper bound solution, however, for prove of concept these numbers were maintained.
(a) (b)
Figure 6-The representation of the number of elements(a) and the σ/τ ratios(b) in the script of the approximate upper bound solution
Subsequently, a python script was used to loop over all stress distributions to find a target shear force and select the distribution with the highest residual moment capacity. The results for the most optimal axial stress distributions are illustrated in Figure 7, and show that the axial stress increased from the neutral line of gravity outwards.
0 0,1 0,2 0,3 0,4 0,5
0 0,2 0,4 0,6 0,8 1
τ/σvM[-]
σ/σvM [-]
-1 -0,5 0 0,5 1
σ/σpl [-]
Height of web divided in 16 elements [-]
V=0,125Vpl V=0,25Vpl V=0,375Vpl V=0,5Vpl V=0,625Vpl V=0,75Vpl V=0,875Vpl
If M-V interaction is determined with a subdivision of areas in shear and axial stress separately, the method is equal to the local approach of (Drucker, 1956) as was discussed in the literature review.
Subsequently a comparison was made between M-V interaction with σ-τ separated or in interaction, corresponding to the local criterion of Drucker (Drucker, 1956) and the coarse upper bound solution obtained from python respectively. At the boundaries, both methods converged to the same solution as illustrated in Figure 8. If the increase in capacity was measured in terms of radii the maximal gain was 19%. However, if the gain was measured in terms of M/Mpl the gain is 220% at V/Vpl=0.875.
Figure 8- MV graph based on σ/τ distributions in separation or in interaction 0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 0,2 0,4 0,6 0,8 1
M/Mpl[-]
V/Vpl [-]
19% Increase σ/τ interaction σ/τ separated
3.2
THEORETICAL APPROXIMATIONStress distributions were used to obtain M-V interaction formula, the shear capacity of the flanges is neglected and therefore an approximation is obtained. First, this approximation was derived analytically. Subsequently, the fillets were added. Addition of the fillets caused a discontinuous width of the web, therefore, this approximation was obtained numerically.
3.2.1 Analytical approximation
As concluded from the literature study; the elastic plastic boundary was used to describe the development of stresses over the web. According to the stress distribution of (Horne, 1951), a formula can be derived. Since the stress distribution of Horne agrees with the equilibrium condition, the formula will as well. The shear stress in the web follows a parabola defined by Jourawski’s Formula (Carpinteri, 1997) :
xy w
V S
t I
(3.1)With
S First moment of inertia
I
Second moment of inertiaV Shear force
tw Thickness of web
xy Shear stressThe maximum shear stress is found at the neutral axis, the derivation is presented in APPENDIX II.
The elastic range of the shear force extends up to 2/3 of
V
pl, subsequently; the shear stress will have an initial value at web boundaries as will be discussed later.2 1 pl 3
z V
V h (3.2)
With
h Height of the web
z1 leading parameter of the elastic plastic boundary
Since this approach is very similar to the one Leth obtained, the outcome is the same interaction plot, as illustrated in APPENDIX II. If the axial stress is assumed to be linear the Von Mises stress will not reach 1 as is illustrated in Figure 9, and similarly proposed by Horne. The increase in shear stress is bounded in z-direction by the equilibrium conditions, however, the axial stress is not. If the Von Mises stress is set to 1, the axial stress can by derived. Both Leth and the proposed solution provided admissible interaction formula below the plastic moment that are suitable for class 3 sections.
Figure 9- Stress distribution over the web by Leth Figure 10- Stress distribution over the web optimized
The formulae for the stresses belonging to Figure 10 were derived, and presented in APPENDIX III.
2
1 1 1
8 3 2 2 4
xz 3
h z z h
h
(3.3)
2 2
2
2 2 4
xx
h z z
h (3.4)The moment is deduced by integrating the stresses over the height of the section
1 2
0
2 ( )
h
M
b z z dz (3.5)Substituting (3.2) and using the maximum section modulus leads to
2 2
0 0
9 9
1 4 16
w w
V V
M M
V V
(3.6)
Since the moment capacity of the flanges may be added, (3.6) can be extended to an interaction formula for the entire section. With
the equivalent of
in the Eurocode.2
0
9 9 1 4 16
analytic
V
V
(3.7)
analytic y pl analytic w
M f W W (3.8)
Equation(3.6) holds until the shear stress reaches the flange-web junctionV V/ pl 2 / 3. It can be assumed that the flanges constrain the shear stress in the web, allowing the shear stress to increase towards0 . However, shear equilibrium at the flange web junction is complicated. A common mistake is to assume the shear stress makes the same jump as the width of the beam. This assumption cannot be true since the bottom surface of the flange should be shear stress free. This means Jourawski’s formula only applies to the extents of the web. The shear flow should always be parallel to the edge’s surface, and cannot be normal to it. The shear flow in an I-beam is depicted in Figure- 11. The shear
stress xz in the web is in equilibrium with
yx in the flange, and explains the change of shear flow direction.Figure- 11- Shear flow Figure 12- Shear stress equilibrium at flange web junction
The shear stress
xy increases linearly from the outer tips of the flanges towards the flange web junction, where it makes equilibrium with xzin the web. Therefore, the shear stress xzin the web needs not be zero at the web boundaries. This stress distribution is supported in literature frequently (Rais-Rohani, 1995) and depicted in Figure 13 .Figure 13- Shear stress distribution
To what extent the shear stress at the web boundaries can increase is dependent on
xyin the flange, and thus dependent on the geometry of the flange. If the flanges are small compared to the web, the flanges might not be able to resist the shear stress of the web. However, the capacity of the flanges is neglected as they are usually little affected. Therefore, it is expected that flanges with insufficient capacity to resist shear stresses at the web junction will not occur.If it is assumed the flanges are capable to resist 0 at the flange web junction, formula (3.6) can be extended to take account of the initial value of xzat the flange web boundaries. The extended version is, however, extensive and added in APPENDIX IV. The extended version holds for V V/ 0 2 / 3, the point where shear stresses gradually increase towards at the web boundaries. The interaction
As expected, the graphs in Figure 14 indicate that the Eurocode M-V interaction EN 1993-1-1, CL. 6.2.8 exceeded the upper bound solution obtained from python. Both the analytical solution and the non- local criterion of Drucker seem to provide an approximation in correspondence with the upper bound solution obtained from python. However, in the literature review it was explained that the non-local criterion by Drucker is not fully applicable to I-shaped sections. In contrary to EN 1993-1-1, CL. 6.2.8, the analytical solution is derived from the equilibrium conditions. Therefore, the analytical solution holds for elastic σ stress distributions, whereas EN 1993-1-1, CL. 6.2.8 needs a plastic hinge. Since the analytical solution is valid for elastic σ stress distributions, it can be used to determine the bending moment capacity for class 3 sections. The analytical solution slightly exceeds the course upper bound solution at V/Vpl=0.875, however, the solution obtained from python could be optimized with a larger range of input variables. Therefore, it was expected the analytical solution would still be below the optimized upper bound solution. The solution by Leth and the local criterion by Drucker are too conservative. Therefore, the analytical solution seems to provide a good alternative to EN 1993-1-1, CL. 6.2.8.
Figure 14- Summary M-V interaction plots from literature, iterative approximation, and analytical solution
3.2.2 Numerical approximation
The influence of the fillets could be added to extend the theoretic approximation. However, the actual influence of fillets was difficult to describe with simple mechanics, as the vector of xz changes direction while the flux is maintained. In this chapter two methods are proposed, in which the vector is remained in plane with the web. Method 1 is based on the principle of a constant shear flow in the web; the shear stress is reduced directly proportional to the increasing width at the fillets. Method 2 uses Jourawski’s formula (3.1) to obtain a shear stress distribution for a discontinuous width.
Incorporating the fillets introduces a discontinuous function for the width. Therefore, both methods will consist of summations for different sections. To apply Jourawski’s formula, the first moment of inertia at the fillets needs to be determined. The first moment of inertia may be defined through equation (3.9).
2
2 2 1
2( ) 3 ( ) cos 2
fillet 4 3
z r r z
S z z r r r h rh h
r z r
(3.9)
Jourawski’s formula is known to be less accurate for non-rectangular shapes (Carpinteri, 1997).
Moreover, a constant shear flow is preferred, therefore method 1 is preferred. Considering approach 1, a decrease in shear stress due to the increasing width allows an increase in axial stress using the Von Mises equation. Formulas are obtained for the development of shear- and axial stresses in the range V/Vpl=0 to 1. These formulas are subdivided to their corresponding sections of the web, as the width is discontinuous. Subsequently, these stresses are integrated over the height of their sections for increasing shear stresses, and create sets of MV values. These sets can be curve fitted to obtain M-V interaction formulas, however, they will be section dependent. The datasets of HEA, HEAA, HEB, HEM and IPE are created with loops in Maple and added in APPENDIX VII, a worksheet for an individual section is added in APPENDIX VI. The datasets indicate 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, as illustrated in Figure 15, seem to indicate that the influence of shear stresses on the moment capacity increases with height. Therefore, M-V interaction design rules might need to incorporate such variables, still minding the usability of the design rule.
4 N UMERICAL MODEL
EQUATION CHAPTER 4 SECTIO N 4To ensure the numerical model accurately describes the mechanical behavior several modelling aspects were evaluated in a parametric study. The influence of the modelling aspects on the behavior of the model were evaluated. Subsequently, to validate the constitutive behavior of the Abaqus model the displacement and rotations were compared to corresponding functions that were derived from mechanics. To extend the validation in plasticity and instability, the numerical model was compared to experimental tests.
4.1
P
ARAMETRIC STUDY 4.1.1 Geometry and setupConsidering the purpose of this research the IPE section was chosen as the basic type section, as it is most prone to instability. During validation, no deviation was made from existing sections. Two lengths were selected to test all aspects for shear dominated(SD)as well as bending dominated(BD) cases. The shear dominated case dimensions were based on the threshold for Euler Bernoulli beam theory.
Eurocode 2-5.3.1(3) does not distinguish aspect ratios for deep beams in 3 or 4-point bending test, instead, it merely states a requirement of 3h between the supports. Therefore, the definition (a=2h) for 4-point bending tests as is illustrated in Figure 16 was obtained from ACI 318 section 10.7. The bending dominated case was defined as (a=9h). During validation, a true stress strain relationship including strain hardening was used as is illustrated in APPENDIX VIII.I.
NEN-EN 1993-1-5.1(2) states webs with hw/t < 72ε should have transverse stiffeners at the supports, since this is well below the slenderness criteria for class 3 illustrated in Figure 2, reference nodes 3/6 and 5/8 were used to model rigid bodies as stiffeners.
The middle lines of the flanges were constrained in X direction, to prevent lateral torsional buckling.
Sections S2 and S3 were used to obtain the bending moment during loading. RP10 was used to read the displacement at mid-span. RP1, RP2 were used to apply a bending moment to the beam, they were coupled to sections S1, S4 respectively through kinematic coupling. RP10 and RP11 were used to apply loads by means of an equation coupling, as will be explained in depth in chapter 4.1.7.
Figure 16-Geometry and setup numerical model parametric study
Furthermore, additional abbreviations were used to address variables in graphs and tables. For instance, the number of elements(EL), force controlled (FC) or displacement controlled(DC), or if half (HALF)a model with symmetry axis was used, otherwise they were specified in the chapter itself.
4.1.2 Elements
According to Dekker, (Dekker, 2018) the fillets influence the shear stress distribution, and
consequently, the shear resisting capacity of steel sections. Therefore, the fillets were not modelled as a simplification of reality and had to be modeled with solid elements. Linear brick elements, except from the C3D8I element, are prone to hourglassing and shearlocking. Therefore, the linear brick elements were compared to the quadratic brick element that should behave well in this situation. The objective was to find an element that behaves well with the least computational power. An overview of the FEM-analyses in this chapter is presented in Table 1, results of the total test population are presented in APPENDIX IX.II.
Table 1- Overview of simulations for evaluating element types
The graphs in APPENDIX IX.II indicate that shear dominated cases were not affected by the type of element, however, bending dominated cases were. Both linear elements C3D8 and C3D8R appeared to be overly stiff in plasticity. Overly stiff behavior was to be expected from the C3D8 element due to shearlocking, the additional capacity while using the C3D8R element in Figure 17 was explained differently. The C3D8R element showed excessive displacements, probably due to hourglassing.
Consequently, large strains correspond to high stresses due to strain hardening. The C3D8R element widely overshoots the plastic bending moment capacity, and thereby confirms strain hardening occurred. Therefore, both C3D8R and C3D8 elements were not suitable elements for the numeric model. The C3D8I and C3D20R element behaved similarly, and accurately obtained the plastic bending moment capacity. Therefore, the C3D8I element was chosen as it required less computational power.
Figure 17- M-δ graphs of 4-point bending test with linear and quadratic elements 0
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6
0 200 400 600 800 1000 1200
M/Mpl[-]
Displacement[mm]
C3D20R-ELEMENT-BD C3D8-ELEMENT-BD C3D8I-ELEMENT-BD C3D8R-ELEMENT-BD
Bending dominated Shear Dominated
C3D8 X X
C3D8R X X
C3D8I X X
C3D20 X
C3D20R X X
4.1.3 Symmetry axis
Addition of a symmetry axes could reduce the size of the model in half, decreasing the required computational power. The symmetry axis was placed at mid span, by constraining translations in Z and X directions. An overview of the FEM-analyses in this chapter is presented in Table 12, the results of the total test population are presented in APPENDIX IX.III.
Table 2- Overview of simulations: Symmetry axis
Without symmetry axis With symmetry axis
FC DC FC DC
4 elements (EL) X X X X
2 elements (EL) X X X X
The buckling mode without symmetry axis indicates the compression flange is angled at mid-span.
However, the symmetry axis pins nodes horizontally and disables rotations as is showed in Figure 18.
(a) Eigenvalue: 376,94 (b) Eigenvalue: 399,8
The symmetry axis affected the buckling modes and increased the eigenvalues, making it an unsuitable method to model a 4-point bending test. The graphs in APPENDIX IX.III support this conclusion. The python input file created an image for every increment that was related to a maximum in the force- displacement(F-δ) graph. The F-δ graphs without symmetry axis as illustrated in Figure 19 , showed two maxima, in which the images clearly showed a first, and second sine wave in the compression flange. These maxima were not present in the F-δ graphs with symmetry axis.
Moreover, the obtained bending moment capacity of the model with symmetry axis was higher, due to the higher eigenmodes. Therefore, a symmetry axis was not applicable in this configuration.
0 10000 20000 30000 40000 50000 60000
0 200 400 600 800 1000 1200
Force[N]
Displacement[mm]
4EL-DC-BD 4EL-FC-BD 4EL-HALF-DC 4EL-HALF-FC Figure 18- Instability with (b) and without (a) symmetry axis
4.1.4 Mesh size section
Prior studies (Dekker, 2018) indicated that shear stresses distributed outwards over the inner sides of the flanges, therefore, a sufficient number of elements had to be used over the thickness of the flanges and web. The number of elements was decreased from 8 to 4, and subsequently 2 as is illustrated in Figure 20. An overview of the FEM-analyses is presented in Table 3, the results of the total test population are presented in APPENDIX IX.IV.
Table 3- Overview simulations: Number of elements over thickness
The computation time of 8 elements BD was excessive, and therefore, this configuration was modeled with a symmetry axis. As explained previously, the validity of the model with symmetry axis expires after reaching Mpl. The graphs in APPENDIX IX.IV, and Figure 21 indicate no conclusive effect of the amount of elements on the results is apparent. The Abaqus manual recommends a minimum of 4 elements over the thickness of plates, therefore, this minimum was adopted for further analyses.
0 10000 20000 30000 40000 50000 60000
0 200 400 600 800 1000 1200
Force[N]
Displacement[mm]
8EL-HALF-DC-BD 2EL-DC-BD 4EL-DC-BD
Bending dominated Shear dominated
FC DC FC DC
2 EL X X X X
4 EL X X X X
8 EL X (HALF) X(HALF) X X
tf: 2 elements tf: 4 elements tf: 8 elements tw: 4 elements tw: 4 elements tw: 8 elements
Figure 20- Mesh study on the amount of elements over thickness of plates
4.1.5 Mesh size aspect ratio
Over the length of shear- axial stress interaction (section a) the aspect ratio was varied. In literature, many different aspect ratio limits are prescribed, however, the correct aspect ratio of an element is very context dependent. Since instability and large deformations were expected, high aspect ratios should be avoided. A well-regarded safe aspect ratio of 1:10 is used in the regions near loads and supports.
Figure 22- regions of the beam with varying mesh size aspect ratios
To avoid erratic behavior the transition of aspect ratios between adjacent regions was smoothened.
The Abaqus manual 6.10 prescribed an aspect ratio limit of 1:20, and therefore, this ratio constituted the limit of this aspect ratio study. Subsequently, the aspect ratio was decreased to 1:10, and 1:5. An overview of the FEM-analyses is presented in Table 4, the results of the total test population are presented in APPENDIX IX.V.
Table 4- Overview simulations: Longitudinal mesh size aspect ratio
Mesh size aspect ratio Bending dominated Shear dominated
1:5 X X
1:10 X X
1:20 X X
The graphs in APPENDIX IX.V, and Figure 23 seem to indicate no conclusive effect of the aspect ratio on the behavior is apparent. Even though the aspect ratio 1:20 seemed to behave well in this configuration, the aspect ratio of 1:10 was adopted to avoid potential problems, while a reasonable computation time was maintained.
0 10000 20000 30000 40000 50000 60000
0 200 400 600 800 1000 1200
Force[N]
Displacement[mm]
1:5-LONG-BD 1:20-LONG-BD 1:10-LONG-BD
4.1.6 Force and displacement controlled Loading
The beam may be loaded by either an imposed displacement or force. An overview of the analyses is presented in Table 5, the results of the total test population are presented in APPENDIX IX.I.
Table 5- Simulation overview: Force controlled or. Displacement controlled
Force controlled or displacement controlled loading should not differ since the RIKS method solves both load and displacement simultaneously by computing the arc length. In contrary to the static general step, forces and displacement are not required to keep increasing while using Riks.
However, as can be seen from the graphs in APPENDIX IX.I, and Figure 24, displacement controlled analyses seemed to be more stable as displacements keep increasing and forces do not. Therefore, displacement controlled analyses were preferred.
Figure 24- F-δ graph of 4-point bending test loaded displacement controlled or force controlled 0
10000 20000 30000 40000 50000 60000
0 200 400 600 800 1000 1200
Force[N]
Displacement[mm]
4EL-DC-BD 4EL-FC-BD
Bending dominated Shear dominated
FC DC FC DC
2 EL X X X X
4 EL X X X X
8 EL X (HALF) X(HALF) X X
4.1.7 Load Application
To apply the load to the beam multiple techniques may be used, the most frequently used in
literature were evaluated in this chapter. The evaluated loadtypes are illustrated in Figure 25, a more elaborate figure is presented in APPENDIX VIII.II. The results of the total test population are
presented in APPENDIX IX.VI. These loadtypes were selected to research the influence of 3 parameters:
-Force controlled or Displacement controlled loading -A load applied to the web or to the flange
-A load applied directly to the section or through a reference point that is tied to that section.
FC-DI-DB FC-RN-DB DC-RN-DB FC-RN-DBLOW
FC-DI-FL FC-RN-FL DC-RN-FL
The influence of the loading on the stress distribution was measured. The stress distribution as a result from loading was similar for all models that were loaded through a reference node. Even loads applied directly to the flange or through a reference node resulted in similar stress distributions as illustrated in Figure 26. Even though the stress distribution of the DBlow model in Figure 26 is less smooth, all models resulted in the same failure loads for shear- as well bending dominated analyses.
Figure 26- shear stress distribution over the width of the loaded flange
The models that were loaded at their flanges showed premature local failure for high shear stresses, as is illustrated in Figure 27. Therefore, the flange imposed loadtypes were excluded from further testing.
Figure 27- Premature local failure of flanges due to imposed loads
The concentrated loads combined with the bending moments were scaled proportionally by the same load proportionality factor (LPF). To obtain results at specific V/Vpl ratios, an estimation had to be made for the required loading. During displacement controlled loading it proved hard to predict the rotations and lateral displacements to apply to obtain results for specific M-V combinations. Therefore, force controlled loading was preferred. As was concluded in chapter 4.1.6, the FC models tended to have convergence issues, however, no convergence issues were encountered with FC-DI DB.
-4 -3 -2 -1 0 1 2 3 4 5
0 50 100 150
τ12[N/mm2]
Width of loaded flange [mm]
FL-DI-FC-BD FL-RN-DC-BD DBLOW-RN-DC-BD
In addition to the loadtype, it was examined if the sequence of loading resulted in order effects. Two methods may be used to obtain the bending moment capacity at a certain shear force:
-Increase the concentrated forces and bending moments simultaneously in a single Riks step
-Increase the concentrated forces up to the desired V/Vpl ratio in a static general step, and subsequently increase the bending moments in a Riks step to determine the residual capacity.
Applying the loads subsequently ensured the bending moment capacity was obtained at the exact intended shear force, however, the results could differ from loading simultaneously as nonlinear material behavior, and instability might cause order effects. Figure 28 indicates no order effects were apparent as both methods obtained the same maxima. However, for high shear forces failure can occur prematurely in the static general step. Subsequently the second step was not initiated and the beam was loaded proportionally nonetheless as is illustrated in Figure 28 . Therefore, all simulations were loaded simultaneously to be consistent.
Figure 28- Comparison of 4-point bending test loaded subsequently(dashed) and simultaneously(continuous) 0
0,2 0,4 0,6 0,8 1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9
M/Mpl[-]
V/Vpl[-]
4.1.8 Imperfections
According to EN-1993-1-5 ANNEX C both geometric and structural imperfections should be included in the FE-model. Imperfections may either be obtained from a refined buckling analysis, or from equivalent values. Any type of imperfection should be taken such that the lowest resistance is obtained. In this section various types of imperfections and methods were evaluated
Geometric imperfection
A geometrical imperfection consists of a combination of an imperfection shape and amplitude. The shape can be equal to one or multiple eigenmodes, or an equivalent shape according to Figure C.1. EN- 1993-1-5 ANNEX C as illustrated in Figure 29. Both methods are used to provide node coordinates of the deflected shape, that are subsequently superimposed on the node coordinates of the GMNIA.
(a) (b)
Figure 29- Imperfections for flange(a) and web(b) according to Table C.2. EN-1993-1-5 ANNEX C
To obtain the eigenmodes of the beam a Linear Buckling Analysis(LBA) with a subspace eigensolver was used. Even though this method is widely adopted, a problem with the LBA was encountered for models loaded in M-V interaction. The graph in Figure 30 indicates an increased bending moment capacity for V/Vpl=0,4 to V/Vpl=0.6. Further inspection into these analyses indicated a discrepancy in the shape between the first 14 eigenmodes and the failure mode in the GMNIA, as illustrated in Figure 32 and Figure 31.
Figure 30- M-V interaction graph with imperfections obtained from a LBA λf =10 λw =83 S235 0
0,2 0,4 0,6 0,8 1 1,2
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
M/Mpl[-]
V/Vpl[-]
Figure 31- First eigenmode of λf =10 λw =83 S235 V/Vpl=0.4 Figure 32- Failure mode of λf =10 λw =83 S235 V/Vpl=0.4
The imperfection obtained from the LBA was not located at the point of failure in the GMNIA; the mismatch resulted in the increased bending moment in Figure 30.
The cause of the mismatch was found when the displacements over the increments of the GMNIA was considered in Figure 33. When stresses are elastic the displacements correspond to the eigenmode of the LBA, subsequently, when the compression flange starts to yield the flange fails premature to buckling of the web. The assumption the member fails in plasticity or elastic buckling seems to be incomplete; elasto-plastic buckling could occur and seems to be sensitive to the applied imperfections.
Obviously, non-linear material behavior is not incorporated in the LBA, and therefore it might provide imperfections in other sections where failure would occur with non-linear material behavior. To check whether the eigenmode is appropriate as input for the imperfection the eigenvalue should be below the elastic moment resistance, however, if not, the researcher cannot switch between methods in one test population.
Increment 9 Mises:
167 N/mm2
Increment 27 Mises:
240 N/mm2 Increment 70 Mises:
242 N/mm2
Figure 33- Evolution of displacements U1 over the increments of the GMNIA
Imperfections should be kept constant to prove causality between slenderness, the influence of shear stress, and moment capacity. This poses another challenge when using eigenmodes as input for the GMNIA analyses. The Abaqus manual recommends applying multiple eigenmodes when the corresponding eigenvalues are close together, however, if the eigenmodes overlap on a single node the imperfections are added up. Therefore, to maintain a constant amplitude the multiple eigenmodes should be selected very carefully. Nonetheless, a constant amplitude for the imperfection for every model is nearly impossible. Even though the LBA is used widely, the difference between linear and non-linear behavior makes it an unsuitable method for many GMNIA analyses.
To obtain the node coordinates of the equivalent shapes a static general step is used. Imposed displacements equal to the amplitude illustrated in Figure 29 were applied to points on the flange and web of the beam.
2 Types of imperfections are applied to the sections in M-V interaction: one shape corresponding to local buckling. one shape corresponding to shear buckling. The stress distribution in a 4-point bending test is identical in both sections, therefore, for any M-V interaction the beam can fail in the lowest failure mode; shear or axial buckling. The amplitude of the imperfection in the outer sections is set to 95% of the middle imperfection, to induce local buckling in the middle and prevent erratic behavior for pure bending.
The buckling shape for shear- and local buckling were recreated from observed failure modes from multiple GMNIA analysis. The double sine is used as buckling shape as it resulted in the lowest bending moment capacity in the GMNIA, as is showed in Table 6.
Table 6- Bending moment capacity for single and double sine imperfection obtained from λf=10 λw=83-S235
Single sine 0.96 M/Mpl
Double sine 0.93 M/MplF
The locations for the points that are displaced were derived from the exact solution for the buckling of members under compression.
Figure 34- Imperfections in Abaqus according to equivalent buckling shapes of Figure 29
The advantage of this method is a constant imperfection for every model in contrary to the LBA, this assures less parameters to consider and a better comparison between the results. Using the equivalent buckling shape, no mismatch was obtained between the applied imperfection and failure mode of the GMNIA. Consequently, no increased bending moment capacity as was illustrated in Figure 30 was recorded. Therefore, the LBA was discarded and the static general step was used to obtain imperfections.
