In 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
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
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
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
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
The moment is deduced by integrating the stresses over the height of the section
Substituting (3.2) and using the maximum section modulus leads to
2 2 formula for the entire section. With
the equivalent of
in the Eurocode.2
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
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