Design of Bending Members in Steel

Steel wide flange beams in an office building

Composite Steel-Concrete Girders

An example of curved steel beams German Historical Museum

Steel Girders for Bridge Decks Sea to sky highway, Squamish

Cantilevered arms for steel pole

What can go wrong ? STEEL BEAMS: Bending failure Lateral torsional buckling Shear failure Bearing failure (web crippling) Excessive deflections

Bending Strength Linear elastic stresses Design Equation: y M Design Equation: Where Fb is the characteristic bending strength For steel this is Fb = Fy For timber it is Fb = fb (KDKHKSbKT)

Plastic moment capacity of steel beams My Fy Mp Fy C T a Ac At Yield moment Plastic moment Which is the definition of the plastic section modulus Z Z can be found by halving the cross-sectional area and multiplying the distance between the centroids of the two areas with one of the areas This is also called the first moment of area So, when do we use the one or the other ??

Steel beam design equation For laterally supported beams (no lateral torsional buckling) Mr =  Fy Z for class 1 and 2 sections Mr =  Fy S for class 3 sections where  = 0.9

Steel cross-section classes Real thin plate sections Will buckle before reaching Fy at extreme fibres Mr < My (Use Cold Formed Section Code S136) Class 3 Fairly thin (slender) flanges and web Will not buckle until reaching Fy in extreme fibres Mr = My Class 2 Stocky plate sections Will not buckle until at least the plastic moment capacity is reached Mr = Mp Class 1 Very stockyplate sections Can be bent beyond Mp and can therefore be used for plastic analysis

Load deflection curves for Class 1 to Class 4 sections

Local buckling of the compression flange

Local torsional buckling of the compression flange

Local web buckling

Lateral torsional buckling Elastic buckling: Mu = ωπ / Le √(GJ EIy ) + (π/L)2EIy ECw Moment gradient factor Torsional stiffness Lateral bending stiffness Warping stiffness Le x y Δx Δy θ

Moment resistance of laterally unsupported steel beams Mr /  Mu Mmax = My for class 3 or Mp for class 1 and 2 0.67Mmax Mmax 1.15 Mmax [1- (0.28Mmax/Mu)] Le

Shear stress in a beam τ τ τmax τmax = V(0.5A)(d/4) ≈ V/Aw (bd3/12)b y A b τ d τmax ≈ V/Aw =V/wd b=w A N.A. y τ d τmax = V(0.5A)(d/4) (bd3/12)b =1.5 V/A

Shear design of a steel I-beam Vr = φ Aw 0.66 Fy for h/w ≤ 1018/√Fy = 54.4 for 350W steel w This is the case for all rolled shapes d h For welded plate girders when h/w ≥ 1018/√Fy the shear stress is reduced to account for buckling of the web (see clause 13.4.1.1) Aw = d.w for rolled shapes and h.w for welded girders

Bearing failures in a steel beam k N N+4t N+10t w Bearing failures in a steel beam For end reactions For interior reactions

Deflections A serviceability criterion Use unfactored loads Avoid damage to cladding etc. (Δ ≤ L/180) Avoid vibrations (Δ ≤ L/360) Aesthetics (Δ ≤ L/240) Use unfactored loads Typically not part of the code Δ