2. Literature survey
2.2 Stress distribution
Stress distributions in steel structures can be based on elastic, elasto-plastic and plastic behavior.
Once the bending moment has a value greater than zero, the elastic deformation starts. Increasing the bending moment will lead to yielding of the steel in the outer fibers though the stress distribution remains elastic. Increasing the bending moment even further will lead to yielding of inner fibers and eventually to yielding of the entire cross-section.
The allowed stress distribution of a cross-section is dependent on the profile classification: Class 1 up to Class 4. This investigation has been limited to Class 1 and 2 of I-shaped cross-sections.
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
13 2.3 Elastic theory
For Class 1 up to Class 4 cross-sections the elastic theory can be used. If the determination of the cross-sectional resistance is based on an elastic stress distribution, the following 3 conditions need to be met according Krachtwerking [3]:
the stress distribution in a cross-section should be in equilibrium with internal forces acting on the cross section;
the yield criterion may not be exceeded at any location in the cross-section;
the stress-strain relation should be linear.
2.3.1 Equilibrium
This condition requires equilibrium of the stresses, which are integrated over the cross-sectional area and the internal forces.
Cross-section subjected to a normal force:
σN,Ed=NEd
A
Cross-section subjected to bending moment:
σM,Ed=MEd
Wel
MEd= σM,Ed∙ Wel (2.2)
Cross-section subjected to a shear force:
τED=VEd ∙ S
MEd design value of the bending moment about the y-axis kNm
Wel elastic section modulus mm³
σM,Ed design value of the normal stresses due to the bending moment N/mm²
VEd design shear force kN
I second moment of area mm4
t plate thickness at location of shear stress determination mm
S first moment of area at that location mm³
τEd design value of the elastic shear stress N/mm² NEd = σN,Ed∙ A (2.1)
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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2.3.2 Yield criterion
The second condition is that the yield criterion may not be exceeded at any location in the cross-section. When regarding an elastic stress distribution for axial stress in one direction, the yield strength should not be exceeded in any fiber. But, when dealing with stress in multiple directions, it gets more complicated. In that case shear stresses may also occur next to the normal stresses, see Figure 2.1.
Figure 2.1 Stresses acting on an infinitesimal fiber particle in (2D) and (3D) [2]
The yield criteria is:
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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2.3.3 Elastic stress distribution
For an elastic stress distribution, the stress-strain relation should be linear, i.e. only the linear elastic part of the steel stress-strain curve may be used, see Figure 2.2 left graph. In this case the relation between the stress and strain is determined by the modulus of elasticity E according to Hooke’s law (σ = E*ε), see left part of Figure 2.2.
Figure 2.2 Linear part stress-strain curve (left), elastic stress distribution by three individual load cases (right) [3]
In case that a structural member is only loaded elastically, the stress distributions are as illustrated by the right picture of Figure 2.2.
2.4 Plastic theory
For Class 1 and 2 sections also the plastic theory can be used. If the determination of the cross-sectional resistance is based on a plastic stress distribution, the following 3 conditions need to be met according to Krachtswerking [3]:
the stress distribution in a cross-section should be in equilibrium with section forces acting on the cross section;
the yield criterion may not be exceeded at any location in the cross-section;
the stresses may be distributed about the cross-section in the most favorable manner, as long as the emerging deformations relate to the plastic stress distribution.
2.4.1 Equilibrium
For any cross-section loaded by an axial force, where equilibrium is met, both the elastic and plastic stress distribution are uniform. Therefore the equation of equilibrium is similar, see equation (2.1).
For a cross-section which is loaded by a bending moment the stresses are uniformly distributed over the section, this results in a higher section modulus (Wpl):
𝜎𝑀,𝐸𝑑=𝑀𝐸𝑑
𝑊𝑝𝑙 𝑀𝐸𝑑 = 𝜎𝑀,𝐸𝑑∙ 𝑊𝑝𝑙 (2.9)
Wpl plastic section modulus mm³
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
16
Equation (2.10) and (2.11) show the plastic section modulus for I-shaped cross-sections regarding the strong and weak axis.
𝑊𝑝𝑙,𝑦 = 𝑏 ∙ 𝑡𝑓(ℎ − 𝑡𝑓) +14∙ 𝑡𝑤(ℎ − 2 ∙ 𝑡𝑓)2 (2.10)
𝑊𝑝𝑙,𝑧=12(𝑏2∙ 𝑡𝑓) +14∙ 𝑡2𝑤(𝑑 − 2 ∙ 𝑡𝑓) (2.11)
b width of a cross-section mm
tf flange thickness mm
h overall depth of a cross-section mm
tw web thickness mm
The plastic moment resistance, Mpl is reached when the yield stress is reached in all fibers. The shape factor α which is the ratio between the plastic and elastic moment resistance, is displayed in equation (2.12).
𝛼 =𝑀𝑝𝑙
𝑀𝑒𝑙 = 𝑊𝑝𝑙
𝑊𝑒𝑙 (2.12)
α shape factor -
Squared section: α = 1.5.
I-shaped cross-section (y-axis): α ≈ 1.15.
In cross-sections loaded by a shear force, the shear stresses should be uniformly distributed over the shear area, see equation (2.13).
𝜏𝐸𝑑 =𝑉𝐴𝐸𝑑
𝑉 𝑉𝐸𝑑 = 𝐴𝑉 ∙ 𝜏𝐸𝑑 (2.13)
𝐴𝑉 shear area mm²
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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2.4.2 Yield criterion
The yield criterion may not be exceeded at any location in the cross-section according Krachtswerking [3]. In the case of a plastic stress distribution the normal stresses are maximum equal to the yield strength. The formulas for the yield criterion as previously described are still valid. The relative reduced yield strength by Von Mises can be written as Equation (2.14).
𝑓𝑦,𝑟 = √𝑓𝑦2− 3𝜏𝐸𝑑2 (2.14)
𝑓𝑦,𝑟 reduced yield strength N/mm²
Rewriting this formula results in equation (2.15).
𝑓𝑦,𝑟
𝑓𝑦 = √1 −3𝜏𝐸𝑑2
𝑓𝑦2 (2.15)
The relative reduced yield strength by Von Mises can be written as equation (2.16). The plastic design shear resistance is equal distributed over the cross-section, see equation (2.17).
𝑓𝑦,𝑟 expressed in the utilization ratio, see equation (2.19).
𝑓𝑦,𝑟
𝑓𝑦 = √1 − 𝑛2 (2.18)
𝑛 = 𝑉𝐸𝑑
𝑉𝑝𝑙,𝑅𝑑 (2.19)
𝑛 utilization ratio -
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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2.4.3 Plastic stress distribution
The last condition states that stresses may be distributed in the most favorable manner within the cross-section, as long as the deformations and plastic stress distribution relate to each other. Figure 2.3 illustrates plasticity in the stress-strain curve by means of a horizontal line, through the yield plateau.
Figure 2.3 Plastic part stress-strain curve [3]
Three cases where single forces are present in the cross-section are displayed in figure 2.4. Every fiber of the cross-section was used, where every fiber reaches the yield strength fy or shear strength τy.
Figure 2.4 Plastic stress distributions by three individual cases [3]
When multiple forces are present instead of a single force, the possible distributions can be shown as figure 2.5. In the case of multiple internal forces the stresses must still remain within the limits of the yield criterion [3].
Figure 2.5 Possible plastic stress distribution in the case of multiple internal forces [3]
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
19 2.5 Classification of cross-sections
NEN-EN 1993-1-1, article 5.5 [1] defines the cross-sectional classes based on the capacity to deform plastically and the ability to redistribute the moments. Figure 2.6 shows the classification in a moment-rotation diagram. Cross-sections will be classified by the c/t ratio of the internal (web) and external (flanges) parts.
t thickness of the flange or web mm
c height of the cross-section minus the flanges and the radii (in case of internal parts) mm
Figure 2.6 Moment rotation behavior for I-shaped beams subjected to bending moment [3]
The four classes of cross-sections are:
- Class 1: plastic sections can form plastic hinges with sufficient rotation capacity to redistribute the moments (strain hardening) before local buckling occurs;
- Class 2: compact sections are able to develop the plastic moment resistance, but the rotation capacity is limited because of local buckling;
- Class 3: semi-compact sections are able to reach the elastic moment resistance, while local buckling prevents the development of the plastic moment resistance;
- Class 4: slender sections are not able to attain the yield stress, because local buckling occurs in one or several parts of the cross-section.
Figure 2.7 shows the different calculation models and corresponding classification of cross-sections.
Figure 2.7 Classification of cross-sections [3]
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
20 2.6 Code requirements
2.6.1 Eurocode 3: EN 1993-1-1
The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules for steel sections are described in NEN-EN 1993-1-1, article 6.2.10 [1].
“Where shear and axial force are present, allowance should be made for the effect of both shear force and axial force on the moment resistance. If the design value of the shear force VEd does not exceed 50% of the design plastic shear resistance Vpl,Rd, no reduction of the resistances defined for bending and axial force need to be made, except where shear buckling reduces the section resistance.
In the case that VEd exceeds 50% of Vpl.Rd, the design resistance of the cross-section to combinations of moment and axial force should be calculated using a reduced yield strength for the shear surface.”
𝑓𝑦,𝑟 = (1 − 𝜌)𝑓𝑦 (2.20) cross-section may be reduced [1].
2.6.2 Dutch National Annex
The Dutch National Annex (NA) of NEN-EN 1993-1-1 article 6.2.10 provides in rules to determine the interaction between bending, shear and normal force. For I-shaped sections of class 1 and 2 the interaction about the strong axis may be checked as follows:
𝑀𝑦,𝐸𝑑
My,Ed design value of the bending moment about the y-axis kNm
My,V,Rd reduced bending resistance due to shear about the y-axis kNm
NEd design value of the axial force kN
NVz,Rd reduced normal force resistance due to shear about the z-axis kN
Npl,Rd design plastic resistance to normal forces of the gross cross-section kN
ᵨ
reduction factor due to shear -Bending moment, shear and normal force interaction of I-shaped steel cross-sections
21
Av shear area mm²
fy yield strength N/mm2
𝛾𝑀0 partial factor for resistance of cross-sections -
2.7 Results from earlier research
2.7.1 Research by Sherbourne & Oostrom, 1972
The investigation of Sherbourne & Oostrom [4] attempted a plastic analysis of castellated beams by developing a computer model which duplicated behavior under incremental, proportional loading.
Moment, shear and axial force interaction surfaces were developed for critical sections of castellated beams and any other I-section beam with a hole in the web. The sections considered the rectangular and T-section. The interaction curves for these shapes were compared with that for the continuous I-section in order to obtain a clearer understanding of the change deriving from a change in shape.
Furthermore, interaction curves were plotted for a complete range of castellated beams in order to assess the effect of changes in size and profile of the web elements and their attendant discontinuities.
Figure 2.8 Castellated beam [4]
To make a 3-dimensional diagram, the shape of the m-v, the m-n and n-v curves as end points were used. The 3-dimensional interaction diagram is shown in Figure 2.9.
Figure 2.9 Geometrical interaction surface [4]
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
22
This research about the castellated beams [4] is not relevant for the current investigation, because old codes where used in the research and the castellated beam differed largely from the I-shaped cross-section. The way of making the 3D-diagram can be used for this research.
2.7.2 Research by Goczek & Supel, 2014
The analytical research of Goczek and Supel [5] includes the resistance of steel cross-sections subjected to bending, shear and axial forces. The aim of their investigation was to analyze the Eurocode 3 rules for the combined bending, shear and axial force interaction. The investigation has been limited to Class 1, 2 and 3 cross-sections. The results showed that the most uncertain design situation arose when bending and axial force were combined with a significant shear force that cannot be neglected. The comparative analysis of interaction diagrams is presented, see Figure 2.10 and 2.11.
Figure 2.10 M-N-V interaction graph – Class 1 and 2 cross-sections[5] Figure 2.11 M-N-V interaction graph – Class 3 cross-sections [5].
The analytical research conducted by Goczek & Supel [5] on the resistance of steel cross-sections subjected to bending, shear and axial forces showed good agreement for Class 1 and 2 cross-sections and poor for Class 3 cross-sections. The pure elastic approach assumed in the Huber-Mises yield criterion, and applied to Class 3 cross-sections excluded partial plastic stress distribution, which is permitted in elastic design by Eurocode 3.
2.7.3 Research by Neal, 1961
Neal, B.G. [6] investigated analytically the effect of shear and normal forces on the fully plastic moment of an I-beam. Therefore a cantilever of I-section subjected to both normal and shear forces at its free end is considered. The results of his investigation are presented in the form of interaction relations between the shear and normal forces and bending moment at the clamped end of the cantilever at failure. The applied forces are shown in Figure 2.12.
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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Figure 2.12 Cantilever I-beam with applied forces [6]
In this research only two cases were investigated, namely neutral axis in the web and neutral axis in the flange. The roots were not taken into acount. The relation between normal force, shear force and bending moment is shown in Figure 2.13.
Figure 2.13 Interaction relations between M-N-V [6]
The two curves which are indicated with (i) and (ii) are from earlier research where bending moment is limited to be zero. This research about normal force, shear force and bending is not relevant for current investigation, because the roots are not taken into account.
2.8 Conclusion regarding literature survey
The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules of M-N-V for steel sections are described in NEN-EN 1993-1-1, clause 6.2.10 [1]. There are two cases:
VEd≤ 0,5 ∙ Vpl,Rd the shear force (V) can be neglected
VEd> 0,5 ∙ Vpl,Rd the design resistance of the cross-section to combination of moment and axial force should be calculated using a reduced yield strength for the shear area
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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3. Analytical solution
In this chapter an analytical solution is presented for a rolled I-shaped cross-section subjected to combined bending, shear force and normal force. The cross-section was divided in several parts to cope with the shear force, normal force and bending moment, see Figure 3.1. The center of the cross-section is reserved for the shear force. The remaining parts of the cross-section are reserved for the normal force and bending moment. Dependent on the amount of applied shear force the size of these areas changes. Figure 3.1 (right picture) shows the geometrical parameters of an I-shaped cross-section.
Figure 3.1 Areas reserved for M-N-V (left) and geometrical parameters of a rolled I-shaped cross-section (right)
Where:
Af area of the flanges (2 ∙ 𝑏 ∙ 𝑡𝑓) mm2
Ar area of the roots (4𝑟2(1 − 0.25𝜋)) mm2
Aw area of the web (𝑡𝑤∙ ℎ𝑤) mm2
As an example an HEA240 cross-section was used to demonstrate the analytical solution. To enable the calculation of M-N-V interaction, three regions and several cases were introduced. Table 3.1 shows an overview of those regions & cases.
Table 3.1 All possible cases for the interaction of M-N-V
When distributing shear force, the regions presented themselves namely:
I) The shear force is only in the web II) The shear force is in the web & roots
III) The shear force is in the web, roots and in the flange
Figure 3.2 shows the three regions where the shear force can be situated.
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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Figure 3.2 Possible positions of the shear force region I, II and III
When distributing the shear force, normal force and bending moment, three general cases presented themselves, namely:
1) The neutral axis of the cross-section is situated in the web 2) The neutral axis of the cross-section is situated in the roots 3) The neutral axis of the cross-section is situated in the flange Three possible cases are shown in Figure 3.3.
Figure 3.3 Possible positions of the neutral axis [7]
Besides these three general cases two subcases emerged, namely 3A and 3B, see Figure 3.4. When the shear force has reached the flange, which is indicated in black in Figure 3.4, then case 3A and 3B would be applied. Flange1 and flange2 in case 3A and 3B are the distances where the normal force can be varied. The difference between the flange and flange1/flange2 is that in case of the flange the shear force is located in web or in the roots and in case of flange1/flange2 the shear force is located in the flange itself. Table 3.1 shows the possible cases.
Figure 3.4 Possible positions of the neutral axis in the flange, Case 3A (left) and Case 3B (right)
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
26 3.1 M-N-V interaction in region I (case 1, 2 and 3)
Case 1: neutral axis in web
In this case, the neutral line is located in the web. As mentioned earlier the shear force is taken by the center part of the web over the height (ℎ𝑦). The normal force is taken by another part of the web which height is indicated with α. The remaining part of the cross-section beared the bending moment, see Figure 3.5.
Figure 3.5 Yield stress distributions in case 1
To determine which part of the cross-section (hy) is reserved for the shear force, equation (3.1) and (3.2) were used.
𝑛𝑉∙𝑉𝑝𝑙
𝑓𝑦⁄√3 = 𝐴 with 𝐴 = 𝑡𝑤∙ ℎ𝑦 (3.1)
ℎ𝑦 =
𝑛𝑉∙𝑉𝑝𝑙 𝑓𝑦 √3⁄
𝑡𝑤 (3.2)
For region I, ℎ𝑦 ≤ 𝑑
Figure 3.6 shows the distances form the center of the cross-section to the center of the gravity of each surface in Case 1.
Figure 3.6 Distance from the center of the cross-section to the center of the gravity of each surface, in Case 1
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
27
In Equations (3.3) to (3.7) the design bending moment with the corresponding plastic section modulus regarding the shear force, normal force and bending moment is described.
𝑊𝑉=1
The design value of the axial force is given by equation (3.10).
𝑁𝐸𝑑 = 𝑓𝑦∙ 𝐴𝑁 = 𝑓𝑦∙ (2 ∙ 𝑡𝑤∙ 𝛼) (3.5)
With:
hv unknown distance in y-direction where shear force is applied mm Wv plastic section modulus regarding the shear force mm³ WN plastic section modulus regarding the normal force mm³ WM plastic section modulus regarding the bending moment mm³
My,Ed design bending moment kNm
In this case the neutral axis is located in the roots. A function is defined to describe the roots, see equation (3.8). This equation can be used to define the area of the roots or to calculate the neutral point of the root, illustrated by Figure 3.7.
(3.8)
Figure 3.7 Area and neutral point of the root [7]
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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Equation (3.8), (3.9) and (3.10) originate from the literature survey of Rombouts, I.M.J. [7].
For the calculation of the root area which is given in equation (3.10), the integral of equation (3.8) is taken.
(3.10)
This results in a slightly different yield stress distribution, see Figure 3.8.
Figure 3.8 Yield tress distributions in case 2
To calculate the plastic section modulus, the distance from the center of the cross-section to the center of gravity of each surface is needed. These distances are shown in Figure 3.9.
Figure 3.9 Distance from the center of the cross-section to the center of the gravity of each surface, in Case 2
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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Equation (3.11) and (3.12) describes the section modulus regarding normal force. In equation (3.13) the normal force is described.
𝑊𝑁= (2 ∙ 𝑡𝑤(𝛽 +𝑑2−ℎ2𝑦)) ∙ (((𝛽+
𝐴𝑟(𝛽) surface area of the roots which is determined by the distance β mm²
In equation (3.14), the section modulus regarding bending moment is described which is used in equation (3.15) to calculate the bending moment.
𝑊𝑀= 𝑊𝑝𝑙− 𝑊𝑉− 𝑊𝑁 (3.14)
Bending moment, shear and normal force interaction of I-shaped steel cross-sections
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Case 3: neutral axis in flange
Case 3: neutral axis in flange