The design of structural steel angles differs from other sections due to the peculiarity of angle sections – their cross-section axes (x-x and y-y axes) differ from their principal axis (u-u and v-v axes) as shown below:
The misalignment between the principal axes and cross-section axes readily makes angle sections to be susceptible to buckling and torsion when loaded. The design of angle section in flexure, compression and tension is discussed in the forthcoming sections.
Design of Steel Angles under Compression when fully restrained
When steel angles are fully restrained and under pure compression, their resistance is like other sections under compression. The expression below has to be satisfied.
$
\frac{N_{ED}}{N_{cRd}}\,\,\leqslant \,\,1
$
NEd is the design compressive force
NcRd is the compression resistance.
i.e: $
N_{cRd}\,\,=\frac{A\,\,f_y}{\gamma _{m0}}\,\,
$ for class 1, 2, & 3 sections
$
N_{cRd}\,\,=\frac{A_eff\,\,f_y}{\gamma _{m0}}\,\,
$ for class 4 sections
Design of Steel Angles under Compression when Unrestrained
When steel angles are under compression and unrestrained, the tendency to buckle should be considered. The estimation of buckling resistance can follow the same methodology as that of other sections under axial compression except for estimating the slenderness of angle sections. According to Annex BB.1.2 of EN 1993-1-1, when angles are subjected to compression such as when in bracings and trusses, provided that the connection is made by at least two bolts and the supporting element provides appropriate end restraint, the eccentricities may be ignored and effective relative slenderness λeff obtained as follows:
λeff,y = 0.5 + 0.7λy
λeff,v = 0.35 + 0.7λv
λeff,z = 0.5 + 0.7λz
The effective slenderness is filled in the expression in estimating the reduction factor for buckling so as to obtain the buckling resistance of angle sections.
Click here to study a worked example of steel angle in 30m span Pratt truss under pure compression
(NB: The angle sections would have been designed as section under combined moment and compression if no adequate restraint is present (such as in the case of a single bolt) to cater for its eccentricity.)
Effective length of angle sections (Leff)
The effective length of angle sections should be taken as the system length except a smaller value can be justified through detailed analysis by the designer.
Design of Steel Angles in Tension
For angles under tension, the expression below must be satisfied
$
\frac{N_{ED}}{N_{tRd}}\,\,\leqslant \,\,1
$
where;
NEd is the design tensile force
NtRd is the tensile resistance. It should be taken as the design plastic resistance of the gross cross-section when the section is without holes.
Angles with welded end connections
For angles with welded connections, the tensile resistance is taken as the design plastic resistance of the gross cross-section.
NtRd = $
N_{plRd}\,\,=\frac{A\,\,f_y}{\gamma _{mo}}\,\,
$
According to clause 4.13 of EN 1993-1-8, an equal angle or unequal angle welded along its larger leg, the effective area is taken as the gross area of the section. For an unequal angle connected by its smaller leg, Anet is taken as the net section area of an equivalent equal angle.
Angles Connected by single rows of bolt through one Leg
When an angle is connected through one leg, there is a localized secondary bending stress in the member due to the eccentricity of the connection. This eccentricity may be ignored – provided that section is connected through single row of bolts – and the angle section designed as concentrically loaded using the below expressions as given in clause 3.10.3 of EN 1993-1-8.
NtRd = Nu,Rd, where the ultimate resistance depends on the number of bolts.
$$
\text{1}bolt:\,\,N_{u,Rd}\,\,=\,\,\frac{\text{2.0\,\,}\left( e_2\,\,-\,\,0.5d_0 \right) \,\,t\,\,f_u}{\gamma _{m2}}
$$
$$
\text{2}bolts:\,\,N_{u,Rd}\,\,=\,\,\frac{\beta _2A_{net}f_u}{\gamma _{m2}}
$$
$$
\text{3}bolts:\,\,N_{u,Rd}\,\,=\,\,\frac{\beta _3A_{net}f_u}{\gamma _{m2}}
$$
Where;
β2 and β3 are reduction factors dependent on the pitch p1 as given in Table 3.8 (reproduced below). For intermediate values of p1 the value of β may be determined by linear interpolation;
Anet is the net area of the angle. For an unequal-leg angle connected by its smaller leg, Anet should be taken as equal to the net section area of an equivalent equal-leg angle of leg size equal to that of the smaller leg.
Design of Restrained Steel Angles in Bending
Design of restrained steel angles in bending is pretty much like any other cross-section profiles. For adequate moment resistance, the equation below should be satisfied.
$
\,\,\frac{M_{Ed}}{M_{cRd}}\,\,\leqslant \,\,1
$
where Mc,Rd is the design bending resistance
$
\,\,Mc,Rd\,\,=\,\,Mpl,Rd\,\,=\,\,\frac{W_{pl}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 1, and class 2 sections)
$
\,\,Mc,Rd\,\,=\,\,Mel,Rd\,\,=\,\,\frac{W_{el,min}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 3 sections)
$
\,\,Mc,Rd\,\,=\,\,\frac{W_{eff,min}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 4 sections)
Design of Unrestrained Steel Angles in bending
When an angle section is unrestrained and also bear loads in bending, it is better off to select an alternative section such as hollow section due to the demand of buckling and torsion. Angle sections are highly susceptible to lateral torsional buckling and there is no guidance in EN 1993-1-1 on how this is to be approached. However, The Institution of Structural Engineers through the publication “Manual for the design of steelwork building structures to Eurocode 3” gives series of expressions that are inspired by BS 5950 to estimate the buckling resistance of angle sections subject to bending. According to the manual, when considering bending of unrestrained rolled angles the moments must be resolved into the directions of the principal axes u-u and v-v. The moment resistance about the v-v axis can be calculated assuming the angle is restrained. The moment resistance about the u-u axis must allow for lateral torsional buckling as follows:
The slenderness of angle section is given in the manual as:
$
\lambda _{LT}\,\,=\,\,0.72V_a\sqrt{\frac{fy}{E}}\phi _a\lambda _v
$
The slenderness λy is that about the minor axis
The equivalent slenderness coefficient Φa is given by the expression:
$$
\phi _a\,\,=\,\,\sqrt{\frac{W_{el,u,\min}^2\,\,g}{A\,\,I_t}}
$$
$$
g\,\,=\,\,\sqrt{\text{1\,\,}-\,\,\frac{I_v}{I_u}}
$$
Va = 1 for equal angles, for unequal angles it is given by the expression below:
$$
V_a\,\,=\,\,\frac{1}{\sqrt{\left( \sqrt{\text{1}+\,\,\left( \frac{4.5\psi a}{\lambda v} \right) \text{2}+\,\,\frac{4.5\psi a}{\lambda v}}\left. \right) \right.}}
$$
References:
EN 1993-1-1: 2005 – Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings
EN 1993-1-8: Design of joints
Manual for the design of steelwork building structures to Eurocode 3
