ΣΦΑΙΡΙΚΗ ΕΙΚΟΝΑ

ΣΕ 1993-1-1: Σχεδιασμός μεταλλικών κατασκευών (Ευρωκώδικας 3) outlines design guidelines for structural steel members for use in buildings using the limit state method. Limit state design entails comparing factored design loads against reduced section and member capacities. These factors are intended to account for variability in loading conditions and material properties. For the ultimate limit state (ULS) design to be satisfied, the following relationship must be true:

\(ULS \;Factor * Load ≤ Reduction \;Factor * Capacity\)

This document outlines the procedure for designing a structural steel member in accordance with EN 1993-1-1 SkyCiv Renderer API ΣΕ 1993-1-1 Σχεδιασμός μελών χάλυβα μονάδα μέτρησης.

Contents

• Ιδιότητες υλικού
  • Fabrication
  • Βαθμός χάλυβα
  • Αντοχή διαρροής
• Section Resistance
  • Κάμψη
  • Κουρεύω
  • Συμπίεση
  • Ενταση
• Member Resistance
  • Κάμψη
  • Συμπίεση

Ιδιότητες υλικού

Fabrication

ΣΕ 1993-1-1 provides design guidance for four types of structural steel fabrication:

• Hot Rolled Sections: Hot rolled sections are manufactured by heating and rolling steel billet through a mill to achieve a required shape. Examples include UB/UC/UBP I-Sections, T-Sections, Channels and Angle sections.
• Welded Sections: Welded (ή κατασκευασμένα) sections are made up of several hot-rolled flat plates welded together longitudinally to form a steel shape. Custom fabricated sections are typically welded.
• Hot Finished Sections: Hot finished sections are produced by heating steel beyond its recrystallisation temperature before rolling to improve the strength of the end product. These sections are almost always structural hollow sections (RHS/SHS/CHS)
• Cold Formed Sections: Cold formed sections are fabricated by pressing steel billet through a mill at room temperature. Cold forming can be used to produce structural hollow sections and thinner open sections. Note EN 1993-1-1 only provides guidance for hollow cold-formed sections.

Βαθμός χάλυβα

Europe and the United Kingdom has numerous steel grades (strengths) that can be used for design in accordance with EN 1993-1-1. There are several European material standards for different types of steel fabrication:

• ΣΕ 10025: Hot rolled products.
• ΣΕ 100210: Hot finished structural hollow sections.
• ΣΕ 10219: Cold formed welded structural hollow sections.

Hot Rolled Sections (ΣΕ 10025)

Common grade availabilities and indicative yield strengths for hot rolled steel shapes are outlined below:

• μικρό 235 (φά_και = 235 MPa)
• μικρό 275 (φά_και = 275 MPa)
• μικρό 355 (φά_και = 355 MPa)
• S450 (φά_και = 440 MPa)

Structural Hollow Sections (ΣΕ 100210 / ΣΕ 10219)

Common grade availabilities and indicative yield strengths for structural hollow sections are outlined below:

• μικρό 235 Η (φά_και = 235 MPa)
• μικρό 275 Η (φά_και = 275 MPa)
• μικρό 355 Η (φά_και = 355 MPa)
• S420 H (φά_και = 420 MPa)
• S460 H (φά_και = 460 MPa)

Αντοχή διαρροής

The yield strength of a material is the stress limit past which plastic deformation will occur. Yield strengths of steel sections are dependent on steel grade and thickness. Typically strength increases with steel grade but decreases with increased steel thickness.

ΣΕ 1993-1-1 Τραπέζι 3.1 provides a simplified approach for calculating the yield strength of a section based on its grade and thickness. A more detailed yield strength calculation can be carried out by referring to the material relevant material standard. The SkyCiv EN 1993-1-1 Steel Member Design module uses the more detailed approach for yield strength calculation.

Selecting a Section in SkyCiv EN 1993-1-1 Σχεδιασμός μελών χάλυβα

The SkyCiv EN 1993-1-1 Steel Member Design tool allows users to select a Standard steel section from the SkyCiv database or design a completely custom section. The program automatically calculates yield strength values for the section flange and web based on the selected steel grade. Users can also adopt a custom steel grade and manually input material properties if required.

 

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Section Resistance

Section Classification

Section Classification is a system used by EN 1993-1-1 to identify the susceptibility of a section to local buckling before attaining its full plastic capacity. Large slender shapes are typically more susceptible to local buckling than small, stocky shapes. Ευρωκώδικας 3 has four Section Classification categories:

• Τάξη 1: ρεplastic moment capacity, meaning the entire section can reach its yield strength under bending and/or compression.
• Τάξη 2:
• Τάξη 3:
• Τάξη 4: Local buckling will occur before yield strength is reached in part of the section.

 

Κάμψη

Section Bending Moment Capacity

ΟΠΩΣ ΚΑΙ 4100:2020 calculates the bending moment capacity of a steel section as follows:

\(M_s = f_y*Z_e\)

Where f_και is the yield stress of the material, and Z_μι is the effective section modulus. The section modulus of a shape is a geometric property which quantifies the bending resistance of a shape. In structural engineering we use two section modulus values, ο elastic (ΜΕ) και πλαστική ύλη (μικρό) συντελεστή ενότητας. Σημείωση, design standards in other regions sometimes interchange the symbols for elastic and plastic section modulus.

The elastic section modulus assumes the entire section (σχήμα) remains elastic under bending, δηλ. no part of the section exceeds the yield strength (φά_και) of the material. This generally occurs when the extreme fibers in the section (top/btm) reach yielding. The elastic section modulus of a section is calculated as follows:

\(Z = \frac{Εγώ}{και}\)

Where I is the second moment of area and y is the geometric centroid of the shape.

The plastic section modulus assumes that the entire section reaches the yield strength of the material under bending, which means parts of the section will exceed the yield strength and experience plastic deformation. The plastic section modulus of a section is calculated as follows:

\(S = A_C*y_C + A_T*y_T \)

Where A_ντο and A_Τ are the areas on either side of the Plastic Neutral Axis (PNA), και γ_ντο / και_τ are the distance from the PNA to the centroid of those areas. Σημείωση, the PNA location is equal to the geometric centroid location for symmetric shapes but will δεν equal the geometric centroid location for asymmetric shapes.

Section Classification

Some steel shapes may have elements of the shape locally buckle before reaching their yield strength, meaning the full capacity of the elastic/plastic section modulus cannot be achieved. This typically occurs in larger, thinner shapes, which are more susceptible to local buckling. ΟΠΩΣ ΚΑΙ 4100 uses the effective section modulus (Ze) value to account for the possibility of local buckling and reduce the bending capacity of the section accordingly. ΟΠΩΣ ΚΑΙ 4100 classifies sections into three categories:

• Compact: Compact sections are not susceptible to local buckling and can achieve their full plastic moment capacity, meaning the entire section can reach its yield strength under bending.
• Non-Compact: Non-compact sections can reach yield strength in the extreme fibres of the section (elastic moment capacity) but cannot achieve their plastic moment capacity before local buckling occurs.
• Slender: Slender sections cannot achieve their elastic moment capacity before local buckling occurs.

Section Slenderness

ΟΠΩΣ ΚΑΙ 4100 determines the section classification by calculating the slenderness of each element within a section and finding the “critical element” that will buckle in compression first. For an I-section, the elements are broken down as shown below. Slenderness values are only calculated for outstand elements, δηλ. elements which are not restrained in both directions. The area of connection between a flange and web (shown in white below) is restrained in both directions and hence is not susceptible to local buckling.

The slenderness of a flat element is calculated as follows:

\(λ_e = \frac{σι}{τ}\τ.μ.{\frac{f_y}{250}}\)

ΟΠΩΣ ΚΑΙ 4100 Τραπέζι 5.2 contains values for plasticity and yield slenderness limits (λ_ep & λ_ey) for compression plate elements based on stress distribution, edge support and residual stresses. The critical element of a section is the element with the highest λ_μι / λ_ey αναλογία. The slenderness values of this element (λ_μι) are used to classify the entire section (referred to as λ_μικρό).

Αν λ_μικρό ≤ λ_sp the section is compact. For compact sections, the effective section modulus is calculated as follows:

\(Z_e = Z_c = min(μικρό,1.5*ΜΕ)\)

Where S is the plastic section modulus, and Z is the elastic section modulus of the section. The term Z_ντο is used interchangeably for the effective section modulus of a compact section.

Αν λ_sp ≤ λ_μικρό ≤ λ_του the section is non-compact. For non-compact sections, the effective section modulus is calculated as follows:

\(Z_e = [(\frac{λ_{του} – λ_{μικρό}}{λ_{του} – λ_{sp}})(Z_c-Z)]\)

Where Z_ντο is the effective section modulus for a compact section.

Αν λ_μικρό > λ_του the section is slender. For a slender section with flat plate elements in uniform compression, the effective section modulus is calculated as follows:

\(Z_e = Z(\frac{λ_{του}}{λ_s})\)

Σημείωση, the effective section modulus for circular hollow sections or flat plat elements with tension at the unsupported edge is calculated differently. Ανατρέξτε στο AS 4100 Ρήτρα 5.2.5 for further information.

Calculating Section Bending Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool calculates slenderness classifications and section bending capacities for positive and negative bending about both principal axes. The results of the slenderness classification check for a 230 PFC are detailed below.

It’s apparent that the slenderness values and section classification are different depending on the direction of bending. This is because the stress distributions and edge support values change depending on which elements are in compression or tension, resulting in different slenderness limit values.

Once section slenderness is known, the module calculates the section bending moment capacity (Κυρία) about each principal axis for positive and negative bending. For symmetric shapes (such as I-Sections), this value will be the same in the positive and negative direction. Asymmetric shapes will have different section bending capacities in the positive and negative bending direction, such as the 230 PFC shown in the example below.

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Κουρεύω

Section Shear Capacity

ΟΠΩΣ ΚΑΙ 4100 only considers the web of a section to contribute to its shear capacity. Hence the shear capacity of a section (Vv) is equal to the web shear capacity. Vertical stiffeners can be added to a section to increase its shear capacity if required. The capacity of an unstiffened web is calculated differently depending on whether the shear stress distribution across the web is uniform or non-uniform. The following shear stress distributions are assumed for standard section shapes:

Uniform Shear Stress Distribution

The shear capacity of a section with uniform shear stress distribution (Β_εσύ) is calculated differently depending on the slenderness of the web panel. For a non-slender web, capacity is calculated as follows:

\(\frac{d_p}{t_w} ≤ \frac{82}{\τ.μ.{\frac{f_y}{250}}}\rightarrow V_u = V_w = 0.6*f_y*A_w\)

For a Circular Hollow Section Vv = V_β = 0.36*f_και*ΕΝΑ (not impacted by section slenderness).

When the web of the section is slender capacity is calculated as follows:

\(\frac{d_p}{t_w} > \frac{82}{\τ.μ.{\frac{f_y}{250}}}\rightarrow V_u = V_b = α_v*V_w\)

\(α_v = \left[\frac{82}{(\frac{d_p}{t_w})\τ.μ.{\frac{f_y}{250}}}\σωστά]^2\)

Where d_Π is the clear depth of the web panel (δηλ. depth excluding flanges), τ_β is the thickness of the web panel, φά_και is the web yield strength and A_β is the gross sectional area of the web. Note A_β is calculated differently for welded and hot rolled sections. For hot rolled sections, ΕΝΑ_β takes the web depth as the entire section depth (ρε). For welded sections, ΕΝΑ_β only takes the clear web depth between the flanges (ρε_Π). Rectangular Hollow Sections also use d_Π for calculation of A_β.

Non-Uniform Shear Stress Distribution

The shear capacity of a section with uniform shear stress distribution (Β_β) is calculated as follows:

\(V_v = \frac{2*V_u}{0.9+\αριστερά(\frac{f*_{vm}}{f*_{va}}\σωστά)} ≤ V_u\)

Where V_εσύ is the section shear capacity with a uniform shear stress distribution and f*_vm /f*_va is the ratio of maximum and average design shear stresses in the web.

Calculating Shear Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool calculates the shear capacity of a section in both principal axes. Minor axis (ΜΕ) shear capacity is calculated using the contribution of the section flanges, excluding any contribution from the section web. Results from the shear capacity calculations for a 200 Μπορείτε να βρείτε μια σειρά τύπων σύνδεσης που ταιριάζουν στις ανάγκες σας χωρίς να αντιμετωπίζετε ακατάστατα συστήματα διεπαφής χρήστη που απαιτούν μια απότομη καμπύλη εκμάθησης για να ξεκινήσετε 22.3 are detailed below.

Συμπίεση

Section Compression Capacity

ΟΠΩΣ ΚΑΙ 4100 calculates the compression capacity (Ν_μικρό) of a concentrically loaded section as follows:

\(N_s = k_f*A_n*f_y\)

Where k_φά is the form factor of the section, ΕΝΑ_ν is the net area of the cross section (gross area excluding penetrations/holes) and f_και is the yield strength of the section. The form factor of a section represents how much of a section can contribute to its compression capacity before local buckling occurs. The form factor is calculated as follows:

\(k_f = \frac{A_e}{A_g}\)

Where A_σολ is the gross area of the section, and A_μι είναι το “effective area” του τμήματος, δηλ. the gross area of the section minus any “ineffective” areas under compression. An ineffective area is part of the section that will buckle before it reaches its yield capacity under compression. Effective areas are calculated by finding the “αποτελεσματικό πλάτος” of each flat plate element within a section and recalculating the section area using these adjusted width values. The effective width of a flat plate element is calculated as follows:

\(b_e = b\left(\frac{λ_{ey}}{λ_{μι}}\σωστά) ≤ b\)

Οπου:

\(λ_e = \frac{σι}{τ}\τ.μ.{\frac{f_y}{250}}\)

Σημείωση, most design software (συμπεριλαμβανομένου SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα) uses the section yield strength for element slenderness calculations, rather than the specific yield strength of the web/flange. This will always provide a conservative result. The b values used for λ_μι calculation are identical to the dimensions used for bending section slenderness checks (with the flange split about the web), but the b used for b_μι calculation is the total flange/web width. λ_ey is taken from AS 4100 Τραπέζι 6.2.4, depending on the edge support and residual stresses of that element.

The effective width of a circular hollow section is calculated as follows:

\(d_e = min(ρε_{ο}\τ.μ.{\αριστερά(\frac{λ_{ey}}{λ_{μι}}\σωστά)}, ρε_{ο}\αριστερά(\frac{3*λ_{ey}}{λ_{μι}}\σωστά)^ 2) ≤ d_{ο}\)

Οπου:

\(λ_e = \left(\frac{d_o}{τ}\σωστά)\αριστερά(\frac{f_y}{250}\σωστά)\)

Calculating Section Compression Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool calculates the form factor and section compression capacity (Ν_μικρό) for standard Australian sections and custom user-defined sections. Results from the section compression capacity calculations for a 610UB 125 are detailed below.

Ενταση

Section Tension Capacity

ΟΠΩΣ ΚΑΙ 4100 calculates the capacity of a tension member (Nt) ως εξής:

\(N_t = min(ΕΝΑ_{σολ}*φά_{και}\; ,\; 0.85*k_t*A_n*f_u)\)

Where A_σολ is the gross area of the section, ΕΝΑ_ν is the net area of the cross section (gross area excluding penetrations/holes), φά_και is the yield strength of the section, φά_εσύ is the tensile (τελικός) strength of the section and k_τ is the tensile force distribution correction factor. The k_τ used in design varies depending on the section shape and connection type. Connections which provide uniform force distribution result in a k_τ factor of 1.0, connections with unequal force distribution result in a k_τ factor between 0.75-1.0.

Calculating Tension Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool allows users to specify the section k_τ value for used in design. A lower k_τ value will result in a lower section tension capacity. Το SkyCiv AS 4100 Member Design calculator assumes no significant holes are present in the section, hence A_ν is taken as equal to A_σολ. Results from the section tension capacity calculations for a 610UB 125 are detailed below.

 

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Member Resistance

Κάμψη

Member Bending Moment Capacity

The bending moment capacity of a steel member may not always be governed by the bending moment capacity of the section (Μ_μικρό). This is because members can fail by another method before the capacity of the section is reached. Lateral torsional buckling is a common failure method for long/unrestrained steel members, which occurs when the section rotates away from its major axis (towards its minor axis) reducing its moment capacity in the direction of bending.

ΟΠΩΣ ΚΑΙ 4100 contains guidance on calculating nominal member capacity (Μ_σι), which factors the section capacity of a steel member (Μ_μικρό) to account for the impact of member slenderness and restraint conditions.

Members with Full Lateral Restraint

Critical Flange

The critical flange of a cross-section is the flange which would deflect the furthest during buckling, eventually resulting in lateral torsional buckling failure. This is typically the compression flange of a member. Critical flange locations for standard sections under vertical loading are shown below.

Full Lateral Restraint

Shorter members with high rotational/lateral rigidity are less likely to rotate out of plane under loading, reducing the likelihood of lateral torsional buckling failure. If a member is short/rigid enough it will be able to reach its section moment capacity (Μ_μικρό) before another failure method occurs. Members which satisfy this condition are deemed as having “Full Lateral Restraint”.

\(Γεμάτος \; Πλευρικός \; θα εκτελέσει τον τύπο σύμφωνα με το AISI S100-12 \; \rightarrow M_b = M_s\)

ΟΠΩΣ ΚΑΙ 4100 Ρήτρα 5.3.2 provides guidance on calculating the Full Lateral Restraint limit for a member. Κυκλικές κοίλες τομές (CHS) and Square Hollow Sections (SHS) are not susceptible to lateral torsional buckling, as they have high lateral/torsional rigidity and equal section moment capacities about both axes. Hence these sections are generally assumed to achieve Full Lateral Restraint regardless of member length.

Continuous Lateral Restraint

Members which have continuous restraint to the critical flange along their entire length are deemed as having “Continuous Lateral Restraint”. Continuous Lateral Restraint is considered equivalent to Full Lateral Restraint for the calculation of member bending capacity (Μ_σι).

Members without Full Lateral Restraint

The bending moment capacity of a member that doesn’t achieve full lateral restraint is calculated as follows:

\(M_b = α_m*α_s*M_s ≤ M_s\)

Where α_Μ is the moment modification factor and α_μικρό is the slenderness reduction factor. ΟΠΩΣ ΚΑΙ 4100 Ρήτρα 5.6 outlines the procedure for calculating α_Μ and α_μικρό.

Minor Axis Member Bending Capacity

The bending capacity for a member bent about its minor axis (Μ_σι) is equal to the minor axis section capacity (Μ_μικρό) about that axis. The minor axis section capacity reflects is the minimum capacity that section can achieve about any axis, hence the member cannot rotate from this axis into a less-favourable orientation.

Calculating Member Bending Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool calculates carries out full lateral restraint checks and calculates member bending moment capacities about both principal axes for positive and negative bending. Users also have the option of selecting “Continuous Lateral Restraint” to bypass the Full Lateral Restraint check. Results from the member bending capacity calculations for a 3m long 200UB22.3 are detailed below.

Σημείωση, this calculator assumes α_Μ = 1.0 and β_μικρό = -1.0 in all calculations. Cantilevered members are not supported by this tool.

Συμπίεση

Member Compression Capacity

The axial compression capacity of a member is also affected by its length, lateral rigidity and restraint conditions. Unrestrained, longer members are likely to fail due to flexural buckling before the section (squash) capacity is reached. ΟΠΩΣ ΚΑΙ 4100 contains guidance on calculating nominal member capacity (Ν_ντο), which factors the compression section capacity (Ν_μικρό) to account for the impact of member slenderness and restraint conditions.

\(N_c = α_c*N_s ≤ N_s\)

Where α_ντο is the member slenderness reduction factor. Ρήτρα 6.3.3 K= Συντελεστής οφέλους δοκιμής 4100 provides guidance on calculating α_ντο. Member compression capacity must be checked about both axes to find the governing value.

Calculating Member Compression Capacity in SkyCiv AS 4100 Σχεδιασμός μελών χάλυβα

ο SkyCiv AS 4100:2020 Σχεδιασμός μελών χάλυβα tool calculates member compression capacity about both principal axes based on restraint lengths and effective length factors specified by the user. Results from the member compression capacity calculations for a 200UB22.3 with an unrestrained length of 4500mm and 1500mm in the Z and Y axis (αντίστοιχα) are detailed below.

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