Civil's Guide
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 steel beams with coccrete 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’d modulus of 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)