Structural Steel, Connections, Bolt Capacities and Effective Lengths
Table of Contents
Structural Steel
Structural Steel is a commonly used construction material which generally forms the ‘skeleton’ of a building. Steel is made by heating iron ore. This is usually done in a blast furnace by blowing oxygen into into the iron and lowering the carbon content (this affects the strength and ductility of the steel). Also, depending on what elements you combine the steel with, will affect its properties(i.e, stainless steel, weathering steel).
Cast iron is an iron-carbon element which contains more than 2% carbon content. This type of material was used in the late 1700s and through the \(18^{th}\) and \(19^{th}\) century during the industrial revolution. However, due to it’s high carbon content, the material is brittle compared to modern day steel. Wrought iron is another material used in the 1900s, but due to it’s brittleness, it was eventually replaced by mild steel. A famous example of a building made of wrought iron is the Effiel tower.
The advantages of using structural steel as a construction material are listed below:
- Strength – high strength/weight ratio
- High aesthetic potential
- Works well with other materials
- Good environment performance – can be recycled or re-used
- Ability to be prefabricated off-site
- Constructed quickly and accurately on site
The disadvantages of using steelwork as a construction material are listed below:
- Needs corrosion protection (and maintenance)
- Cost of steelwork may vary (depends on international price as RC frames may sometimes be cheaper)
- Flexibility and stability (stability usually achieved through steel bracing, but bracing locations will most likley have issues with architects plans and layouts)
- Vibration – especially for hospitals, stadiums and gyms
Hot Rolled Steel
Hot rolled steel is formed in a mill process by rolling steel at a high temperature of up to \(1700^\circ\). The steel can then be shaped and formed easily. When the steel cools off, it will slighly shrink which results in less control over the final shape (i.e, check allowable tolerances). Most steel frames are designed with hot-rolled steel members.
Cold Rolled Steel
Cold rolled steel involves an additional process ater hot rolling, where the material is cooled, followed by annealing (heat treatment to remove internal stresses and toughen it. This increases the yield stress but comes at the expense of ductility and toughness. General materials which are cold rolled are purlins on a roof.
Steel material properties
| Density | 78.5 \(kN/m^3\) |
| Tensile Strength | 275-460 \(N/mm^2\) yield stress and 430-550 \(N/mm^2\) ultimate strength |
| Poisson’s ratio | 0.3 |
| Modulus of elasticity, E | 200 – 210 \(kN/mm^2\) (GPa) |
| Modulus of rigidity, G | 79 – 81 \(kN/mm^2\) (GPa) |
| Linear coefficient of thermal expansion | 12 x \(10^{-6/\circ}C\) |
Steel Durability and Exposure Conditions
Durability of Steel
Over the lifetime of a structural steel frame structure, the members will start rusting over time. This is due to the presence of water and oxygen in the air, which causes a
chemical reaction on the surface of the steel producing rust \(Fe_2O_3\). The rate of corrosion depends on the atmospheric pollution (i.e, there is a lot of cholride present
in costal and marine areas which will accelerate the corrosion rate).
When designing any steel structure and it’s elements, it is important to take into consideration it’s durability aspect and location within the country.
Keep in mind the types of protective measures that can be used for steel (i.e, paint coatings, galvanised steel, weathering steel and stainless steel). In addition,
bi-metallic corrosion can be issue at connections where you have 2 different types of metals in contact with each other.
Corrosion and Fire protection
Corrosion and fire protection is achieved through paint coatings, galvanised steel and alloys.
This gives structural steel protection as required.
Paint Coatings
When applying paint coatings on a structural steel member, this creates a barrier between the surface of the member and atmosphere, so water and oxygen does not penetrate causing corrosion. When applying paint coatings, the surface needs to be cleaned (achieved through blast cleaning), before applying a primer coat (wet and adheres to substrate and acts as inhibitor), and then apply an intermediate coats (builds film thickness), before applying the finishing coat (Resistance to the environment and provides aesthetics).
The paint is generally applied via brush, roller and sprayed. This can either be done in a shop before transported to site or site applied. It should be considered that shop applied coats can be easily damaged during transportation (barrier is broken) and needs to be rectified on site.
Galvanised Steel
Galvanised steel is the process of applying zinc coating to the steel. This coating acts as a barrier between the steel and atmosphere. Initially, the steel member needs to be cleaned and treated. The steel is immersed in a fluz solution, which helps prevent further oxidation before galvanising begins. The steel is then dipped in a bath of molten zinc, which produces a zinc-iron alloy. This process is can be done for strucural steel members and small components such as fasteners.
Weathering Steel
Weathering steel is a high strength, low alloy steel which forms an adherent protective rust ‘patina’ that reduces corrosion. Cor-ten is the name given to weathering steel and the low rate of corrosion can mitigate the need for protecttive painting and achieve a high design life (120 years for bridges) with minimal maintenance. The standard for weathering steel is located in BS EN 10155. The benifits of weathering steel are listed below:
Durability Exposure conditions
The tables below defines the ‘environment category’ which is based on BS EN 12944-2 and BS EN ISO 9223. The corrosivity class should be defined during the start of your project (i.e, based on location of building)
Structural Steel Connections
Types of Connections
Connections can be defined as rigid, semi-rigid or flexible which allows rotation. The type of connection chosen will affect the analysis of a steel member.
A rigid and semi-rigid connection can transfer moment between Structural steel members. An example is a portal frame structure, which commonly uses haunches (rigid joints) with
end plates which gives fixity between the column and beam allowing the transfer of moment.
Flexible connections do not resist any moments and allows rotation, which is usually analysed as pin support connection. There will always be a small amount
of moment transfered but this is a neglible amount and assumed to be zero. In construction, this is achieved through using fin plates and flexible end plates. The plates
are usually welded to a column or welded to another beam and connected via bolts. However, the bolts will not provide any resistance to rotation.
In general, beams are designed as simply supported (flexible connection) as moment connections can become expensive, especially complex connections. Moment connections
provides stability to a structure, but stability can be achieved via cross bracing. The use of cross-bracing in a design can help avoid the use of these complex connections.
Overall, the engineer/designer needs to determine the nature of the connection, is it a flexible connection or moment connection?
Welded connections
| Leg Length s (mm) | Throat thickness a = 0.7s (mm) | Longitudinal capacity \(F_{W,L},R_s\) (kN/mm) | Transverse capacity \(F_{W,T},R_s\) (kN/mm) |
|---|---|---|---|
| 4 | 2.8 | 0.7 | 0.88 |
| 5 | 3.5 | 0.88 | 1.09 |
| 6 | 4.2 | 1.05 | 1.31 |
| 8 | 5.6 | 1.4 | 1.75 |
| 10 | 7.0 | 1.75 | 2.19 |
| 12 | 8.4 | 2.10 | 2.62 |
| 15 | 10.5 | 2.62 | 3.28 |
Refer tata steel blue blook on fillet welds
\(F_{w,Ed}\leq F_{w,Rd}\)
where \(F_w,Ed\) is the design value of the weld force per unit length and \(F_w,Rd\) is the design weld resistance per unit length
\(F_w,Rd=f_{vw.d}a\)
\(f_vw.d\) is the design shear strnegth of the weld
\(f_vw.d=\frac{f_u/\sqrt{3}}{\beta_w\gamma_{M2}}\)
\(\beta_w = 0.9\), \(\gamma_{m2}=1.25\) and \(f_u = 410 N/mm^2\)
Welding is used to join structural steel members together via a process called metal arc welding. There are many different welding techniques but the welding
process uses an electric arc to generate heat which melts the base metal. A seperate filler material is also used help this process to joining the metal pieces together.
The arc generates heat and the filler material is usually welding wire or stick electrodes.
Bolt capacities for non-preloaded bolts - Class 8.8 hexagon head bolts with S355
| Diameter of bolt, d (mm) | Tensile stress area \(A_s\) \(mm^2\) | Tension resistance \(F_t,Rd\)(kN) | Shear resistance | Bolts in tension \(t_{min}\) (mm) | |
|---|---|---|---|---|---|
| Single Shear \(F_{v,Rd}\)(kN) | Double Shear \(2\times F_{v,Rd}\) (kN) | Min. thickness for punching shear | |||
| 12 | 84.3 | 48.6 | 27.5 | 55.0 | 3.7 |
| 16 | 157 | 90.4 | 60.3 | 121 | 5.5 |
| 20 | 245 | 141 | 94.1 | 188 | 6.8 |
| 24 | 353 | 203 | 136 | 271 | 8.2 |
| 30 | 561 | 323 | 215 | 431 | 10.1 |
Bolted Connections
End plate connections are a common design in steel structures and invloves using a steel plate connected to a structural steel member via bolts. There are different Types
of bolts and classification which affect its strength and properties. The most common bolts used are hexagonal head bolts and countersunk bolts.
There are minimum and maxximum spacing requirements as mentioned in BS EN 1993-1-8.
| Distances and spacing | Minimum | Maximum | |
|---|---|---|---|
| steel exposed to the weather or other corrosive influneces | Steel not exposed to the weather or other corrosive influences | ||
| Edge distance \(e_1\) | 1.2\(d_0\) | 4t+40mm | |
| Edge distance \(e_2\) | 1.2\(d_0\) | 4t+40mm | |
| Spacing \(p_1\) | 2.2\(d_0\) | The smaller of 14t or 200mm | The smaller of 14t or 200mm |
| Spacing \(p_2\) | 2.4\(d_0\) | The smaller of 14t or 200mm | The smaller of 14t or 200mm |
\(d_0\) = Hole diameter
The table above is table 3.3 from BS EN 1993-1-8
A clearance holes is a hole through a material which is oversized so the threads of a screw or bolt can pass through but not the head. For bolts
which are below 24mm, a 2mm clearance hole is required. For exmaple, an M12 bolt will require a 14mm clearance hole. Any bolts larger than 24mm will require a
3mm clearance hole.
Bolt capacities for non-preloaded bolts - Class 8.8 countersunk bolts with S355
| Diameter of bolt, d (mm) | Tensile stress area \(A_s\) \(mm^2\) | Tension resistance \(F_t,Rd\)(kN) | Shear resistance | Bolts in tension \(t_{min}\) (mm) | |
|---|---|---|---|---|---|
| Single Shear \(F_{v,Rd}\)(kN) | Double Shear \(2\times F_{v,Rd}\) (kN) | Min. thickness for punching shear | |||
| 12 | 84.3 | 34.0 | 27.5 | 55.0 | 2.6 |
| 16 | 157 | 63.3 | 60.3 | 121 | 3.8 |
| 20 | 245 | 98.8 | 94.1 | 188 | 4.8 |
| 24 | 353 | 142 | 136 | 271 | 5.7 |
| 30 | 561 | 226 | 215 | 431 | 7.1 |
Effective Lengths and Preliminary Sizing
Effective Length
Effective length factors are used in the design of structural steel members and factors correspond on the condition of the beam during construction and in its permanent state. For example, a common example in the design of a beam is checking the top flange fully restrained (usually by floor plate/slab) or unrestrained.
The effective lentgh factors are tabulated below:
D is the overall depth of the beamThis extract is table 13 in BS 5950-1
| Conditions of restraint at supports | Loading conditions | ||
|---|---|---|---|
| Normal loads | Destabalising loads | ||
| Compression flange laterally restrained; beam fully restrained against torsion (rotation about the longitudinal axis) | Both flanges fully restrained against rotation on plan | 0.7L | 0.85L |
| Compression flange laterally restrained; beam fully restrained against torsion (rotation about the longitudinal axis) | Compression flange fully restained against rotation on plan | 0.75L | 0.9L |
| Compression flange laterally restrained; beam fully restrained against torsion (rotation about the longitudinal axis) | Both flanges partially restrained against rotation on plan | 0.8L | 0.95L |
| Compression flange partially restrained against rotation on plan | 0.85L | 1.0L | |
| Both flanges free to rotate on plan | 1.0L | 1.2L | |
| Compression flange laterally unrestrained. Both flanges free to rotate on plan. | Partial torsional restraint against rotation about longitudinal axis provided by connection of bottom flange to supports | 1.0L + 2D | 1.2L + 2D |
| Compression flange laterally unrestrained. Both flanges free to rotate on plan. | Partial torsional restraint against rotation about longitudinal axis provided only by pressure of bottom flange onto supports | 1.2L + 2D | 1.4L + 2D |
Effective Length of cantilevers without intermediate restraint
This extract is table 14 in BS 5950-1
| Restraint conditions | Effective Length | ||
|---|---|---|---|
| Support | Cantilever Tip | Normal load | Destabalising loads |
| Continuous with lateral restraint to top flange |
|
|
|
| Continuous with partial torsional restraint |
|
|
|
| Continuous with lateral and torsional restraint |
|
|
|
| Restrained laterally, torsionally and against rotation on plan |
|
|
|
Preliminary Sizing of Steel Beams
When starting the design of any stuctural members, there are rules of thumb (span to depth ratio) which are commonly used to determine the size of a members
which provide a starting point in a design before refining it through calculations and checks.
When structural steel beams depths become large/long spans, deflection will govern the design
| Floor Type | Span (m) | |
|---|---|---|
| Composite beam & in-situ composite slab | 6-12 | |
| Steel beam & precast slab | 6-9 | |
| Slimfor beam & precast slab | 6-12 | |
| Composite beams with web openings | 6-12 | |
| Castellated/Cellular beams | 6-12 | |
| Truss | 6-12 | |
| Composite plate girder | 6-12 | |
| Element | Depth | |
|---|---|---|
| Non-composite primary beams | Floor = span/20 | |
| Roof = span/25 | ||
| Non-composite secondary beams | Floor = span/25 | |
| Roof = span/30 | ||
| Composite beams |
| |
Typical Column sizes
The maximum length of a column is dictated by the transportable length on the back of a truck/wagon on the motorway. The maximum lnegth is typically 12m – 15m before a splice connection between 2 columns.
| Number of floors supported by column | Universal column size (UC) |
|---|---|
| 1-2 stories | 152 UC |
| 2-4 stories | 203 UC |
| 3-6 stories | 254 UC |
| 5-12 stories | 305 UC |
| 8-12+ stories | 356 UC |
Section classification of Structural Steel
Structural steel sections are classified as either elastic or plastic which is determined in accordance to Eurocde 3 (BS EN 1993).
The thinner the web and plate sections in steel means that the beam will likely buckle locally as the section is slender.
This means the section cannot achieve its full plastic capacity. This will impact the capacity of the steel in design scenarios.
British standards used to refer the classifications as plastic, compact, semi-compact and slender before changing to class 1,2,3 and 4 in accordance to Eurocodes.
Before beginning the design of any structural steel member, the section classification must be confirmed, which will allow us to determine it’s capacity when checking against
bending, shear, etc.
| Cross-section classification | Method of global analysis | Method of section analysis | Description |
|---|---|---|---|
| Class 1 | Plastic | Plastic | achieves a plastic hinge and has rotatational capacity required for plastic analysis. |
| Class 2 | Elastic | Plastic | achieves a hinge, but limited rotational capacity due to local buckling. |
| Class 3 | Elastic | Elastic | achieves yield strength with elastic distribution, but local buckling prevents development of the plastic moment resistance. |
| Class 4 | Elastic | Elastic plate buckling | local buckling prevents attainment of yield |
Steel Design Equations
Partial Saftey Factors
When designing in accordance to Eurocode 3 (BS EN 1993) for steel, partial factors, \(\gamma_M\) is applied to the characteristic values of resistance (i.e, material properties to take into account manufactoring errors)
| Resistance of section | Description | Partial factor value | ||
|---|---|---|---|---|
| \(\gamma_{M0}\) | resistance of cross-sections whatever the class is: | 1.0 | 0.85L | |
| \(\gamma_{M1}\) | resistance of members to instability assessed by member checks: | 1.0 | ||
| \(\gamma_{M2}\) | resistance of cross-sections in tension to fracture: | 1.25 (1.10 to Bristish National Annex) | ||
| Joints | resistance of joints: | 1.25 | ||
Refer to BS EN 1993-1-1, section 6.1
Tension
The design value for tension is denoted by \(N_{ED}\) and should satisfy the equation below:
\(\frac{N_{ED}}{N_{t,Rd}}\leq 1.0\) Equation 6.5 of BS EN 1993-1-1
The design resistance \(N_{t,Rd}\) should be taken as the smaller of the following equations below:
| \(N_{pl,Rd} = Af_y/\gamma_{M0}\) | design plastic resisance of the gross cross-section |
| \(N_{u,Rd} = 0.9A_{net}f_u/\gamma_{M2}\) | design ultimate resisance of the net corss-section at holes for fasteners |
Compression
The design value for compresison is denoted by \(N_{ED}\) and should satisfy the equation below:
\(\frac{N_{ED}}{N_{c,Rd}}\leq 1.0\) Equation 6.5 of BS EN 1993-1-1
The design resistance \(N_{t,Rd}\) should be taken as the smaller of the following equations below:
Refer to section 6.2.4 of BS EN 1993-1-1
| \(N_{c,Rd} = Af_y/\gamma_{M0}\) | for class 1,2 or 3 cross setions |
| \(N_{c,Rd} = A_{eff}f_y/\gamma_{M0}\) | for class 4 cross-sections |
In my personal experience, avoid using class 3 and 4 classification when designing steel members.
Bending moment
The design value of the bending moment \(M_{Ed}\) at each cross-section shall satisfy equation 6.12 of BS EN 1993-1-1.
\(\frac{M_{Ed}}{M_{c,Rd}\leq 1.0\)
The design resistance for bending about one principle axis of a cross section is shown below for the following classes.
| Design resistance to bending | Section classificatio |
|---|---|
| \(M_{c,Rd} = M_{pl,Rd} = \frac{W_{pl}f_y}{\gamma_{M0}}\) | For class 1 or 2 cross sections |
| \(M_{c,Rd} = M_{el,Rd} = \frac{W_{el,min}f_y}{\gamma_{M0}}\) | For class 3 cross sections |
| \(M_{c,Rd} = \frac{W_{eff,min}f_y}{\gamma_{M0}}\) | For class 4 cross sections |
However, for class sections 1 or 2 that undertakes high shear, (\(V_{c,Ed} \geq 0.5 V_{c,Rd}\)), \(M_{c,Rd}\) is typically
\(M_{y,V,Rd} = \frac{1}{\gamma_{m0}}[W_{pl}-\rho \frac{A^2_w}{4t_w}]f_y\)
\(A_w\) refers to the area of the web in I-sections
\(\rho = ((2V_{Ed}/V_{pl,Ed})-1)^2\) and \(W_{pl}\) is the plastic modulus of the section
It is worth designing a steel member with section classification of 1 or 2, and avoid class 3 or 4 sections as the analysis starts to become more complex, when
it is easier to use a heavier section which can be the difference between class 2 and 3.
\(f_y\) is the yield strength of steel. In the UK, it is common to use S355 (structural steel grade) which has an \(f_y = 355 N/mm^2\)
The moment capacity of a capacity \(M_c,Rd\) is also affected by the restraint conditions of the compression flange. During the construction of a typical steel structure, using structural steel beams with concrete decking connected via a metal deck and shear studs, the top flange will be fully restrained. If you look at the bending moment of a simple supported beam, there a is a sagging moment (tension in the bottom flange and compression in top), and the concrete deck restrains the top flange (i.e. Full restraint) of top flange).
Note – When looking at cantilever or continuous beams with a free end; there will be a reversal of moments (i.e. compression flange will be along the bottom flange of an I-section), the compression flange becomes unrestrained and this reduces moment capacity of the section (buckling length of the section becomes full length of the beam). The reduced bending moment capacity is defined as: \(M_{b,Rd}\).
Shear
The design shear force \(V_{Ed}\) (applied shear force on a cross section) must satisfy the expression below:
\(\frac{V_{Ed}}{V_{c,Rd}}\leq 1.0\)
The design plastic shear resistance is defined by the equation below:
\(V_{c,Rd} = \frac{A_v(f_y/ \sqrt {3})}{\gamma_M0}\) where \(A_v\) is the shear area
| Shear Area | Section Types |
|---|---|
| Rolled I and H destions, load parallel to web | \(A-2bt_f+(t_w+2r)t_f\) but not less than \(\eta h_w t_w\) |
| rolled channel sections, load parallel to web | \(A-2bt_f+(t_w+r)t_f\) |
| rolled T -section, load parallel to web | \(A-bt_f+(t_w+2r)\frac{t_f}{2}\) |
| for welded T -section, load parallel to web | \(t_w(h-\frac{t_f}{2})\) |
| welded I,H and box sections, load parallel to web | \(\eta\Sigma(h_wt_w\) |
| welded I, H, channel and box sections, load parallel to flanges | \(A-\Sigma(h_wt_w\) |
| rolled rectangular hollow sections of uniform thickness with load parallel to depth | \(Ah/(b+h)\) |
| rolled rectangular hollow sections of uniform thickness with load parallel to width | \(Ab/(b+h)\) |
| circular hollow sections and tubes of uniform thickness | \(2A/\pi\) |
| Notation | |
|---|---|
| A | Cross sectional Area |
| b | overall breadth |
| h | overall depth |
| \(h_w\) | depth of the web |
| r | root radius |
| \(t_f\) | flange thickness |
| \(t_w\) | web thickness (if web thickness is not constant, than take minimum thickness) |
| \(\eta\) | may taken as 1.0 (conservative) |
Refer to BS EN 1993-1-1, section 6.2.6 (Shear)
Lateral Torsional Buckling
As mentioned previously, the reduced moment capacity, \(M_{b,Rd}\) depends on the slenderness of the section (i.e, compression flange is not restrained).
The buckling factor \(\chi_{LT}\) and is used to determine the reduced capacity.
\(\frac{M_{Ed}}{M_{b,Rd}}\leq 1.0\)
\(M_{b,Rd} = \chi_{LT}W_y\frac{f_y}{\gamma_M1}\)
| \(W_y = W_{pl,y}\) | for Class 1 or 2 cross-sections |
| \(W_y = W_{el,y}\) | for Class 3 cross-sections |
| \(W_y = W_{eff,y}\) | for Class 4 cross-sections |
\(\chi_{LT} = \frac{1}{\Phi + \sqrt {\Phi^2_{LT} – \bar{\lambda}^2_{LT}}}\) but \(\chi_{LT} \leq 1.0\)
Buckling paraemeter = \(\Phi = 0.5[1+\alpha_{LT}(\bar{\lambda}_{LT}-0.2)+\bar{\lambda}^2_{LT}]\)
where \(\alpha_{LT}\) is an imperfection factor
\(\bar{\lambda}_{LT} = \sqrt{\frac {W_yf_y}{M_{cr}}}\)
The imperfection factors \(\alpha_{LT}\) can be found in table 6.3 in BS EN 1993-1-1. (Use appropriate National annex)
| Buckling curve | Imperfection factor |
|---|---|
| a | 0.21 |
| b | 0.34 |
| c | 0.49 |
| d | 0.76 |
| Cross-section | Limits | Buckling Curve |
|---|---|---|
| Rolled I-sections | \(h/b\leq2\) | a |
| \(h/b\geq2\) | b | |
| Welded I-sections | \(h/b\leq2\) | c |
| \(h/b\geq2\) | d | |
| Other cross-sections | \(h/b\geq2\) | d |
For slenderness, \(\lambda_{LT} \leq \lambda_{LT,0}\), lateral torsional buckling effects may be ignored and only cross sectional checks apply.
\(M_{cr}\) is the elastic critical moment for lateral-torsional buckling. This value is quite long-winded to calculate and BS EN 1993-1-1 does
not really give any adequate guidance on it’s calculation. This is generally calculated using NCCI documents (Non-contradictory complementary infomation). The
document is called SN003b and can be easily found online.
\(M_{CR}=\frac{C_1xEI_z}{L^2}[\sqrt{\frac{I_w}{I_z}+\frac{L^2GI_w}{\pi^2EI_z}+(C_2Z_g)} – C_2Z_g]\)
\(I_z\) is the minor axis second moment of area, E is the Young’s modulus of structural steel \(210 N/mm^2\), \(I_T\) is the warping constant. These values can be
found using the online tata steel blue book resource which contains these values for I-sections.
\(C_1\) and \(C_2\) are factors that considers support conditions, section properties and destabilising loads. For a beam without destabilising loads, \(C_2Z_g = 0\).
The values of \(C_1\) can be obtained from SN003b.
Combined bending and axial Check
Sections which undertake bending moment and axial force will need to be checked against section 6.3.3 of BS EN 1993-1-1. An example is a column subjected to wind loading (bending moment formed in one axis) and subjected to axial loads from upper floors
The simplified expression below can be used for the design of columns in these various scenarios:
- The column is a hot rolled I, H or rectangular section
- Section is classification is 1,2 or 3
- Bending moment about each axis are linear
- Column is retrained laterally at each floor but unrestrained between floors (i.e, restrained top and bottom nodes but flanges are unrestrained)
\(\frac{N_{Ed}}{N_{min,b,Rd}}+\frac{M_{y,Ed}}{M_{y,b,Rd}}+1.5\frac{M_{z,Ed}}{M_{z,cb,Rd}}\leq 1.0\)
\(N_{Ed},M_{y,Ed},M_{z,Ed}\) can be found in BS EN 1993-1-1
\(N_{min,b,Rd}\) is the lesser of \(\frac{\chi_y f_yA}{\gamma_{M1}}\) and \(\frac{\chi_z f_yA}{\gamma_{M1}}\)
\(M_{y,b,Rd}\) is equal to \(\chi_{LT}\frac{f_yW_{pl}}{\gamma_{M1}}\)
\(M_{z,cb,Rd}\) is given by \(M_{z,cb,Rd}=\frac{f_yW_{pl}}{\gamma_{M1}}\), which is the same as the bending resistance of the cross section \(M_{c,Rd}(=\frac{f_yW_{pl}}{\gamma_{M1}})\)
The check above is a simplification of a check found in EN 1993-1-1, which are expressions 6.61 and 6.62. When designing members, either the above equation needs to be checked or expressions 6.61 an 6.62.
\(\frac{N_{Ed}}{N_{y,b,Rd}}+k_{yy}\frac{M_{y,Ed}}{M_{y,b,Rd}}+k_{yz}\frac{M_{z,Ed}}{M_{z,cb,Rd}}\leq 1.0\) (ex. 6.61)
\(\frac{N_{Ed}}{N_{z,b,Rd}}+k_{zy}\frac{M_{y,Ed}}{M_{y,b,Rd}}+k_{zz}\frac{M_{z,Ed}}{M_{z,cb,Rd}}\leq 1.0\) (ex. 6.62)