CTC 422 Design of Steel Structures Beams Shear and Deflection

Beam Design Student Objectives Analyze a beam to calculate load, shear, moment and deflection and to determine if a given beam is adequate Design (select) a beam to safely to support a load considering moment, shear and deflection

Design for Shear - LRFD Steel beams  Shear seldom governs Exceptions: Very short spans Very heavy loads Normal procedure: Select beam for moment (bending) Then check for shear

Load and Resistance Factor Design - LRFD Design strength ≥ Required strength ΦRn ≥ Ru For shear Φv Vn ≥ Vu Where: Vn = Nominal shear strength Φv = Strength reduction factor for shear Vu = Required shear strength based on factored loads

Design for Shear Chapter G of AISC Code Nominal shear strength, Vn, depends on the failure mechanism of the beam Beam can fail by: Shear yielding Shear buckling Can be inelastic or elastic buckling Failure mechanism is related to the width to thickness ratio of the beam’s web, h / tw Section G 2.1 applies to doubly symmetric sections (W, M, S and HP shapes) and channel shapes (C and MC) Section G 2.1a applies to I-shaped sections with h / tw ≤ a given limit Section G 2.1b applies to other doubly symmetric shapes and channels Nominal shear strength in this case is less than calculated by G 2.1a

Design for Shear Chapter G of AISC Code Nominal shear strength, Vn, can be calculate by Section G 2.1a, or G 2.1b Then check ΦvVn ≥ Vu Or, shear strength ΦvVn tabulated in design aids can be used Shear strength is listed in the following tables: W-Shapes – Table 3-2 Also listed in Table 3-6 S-Shapes – Table 3-7 Channel Shapes (C and MC) – Tables 3-7 and 3-8

Seviceability Chapter L of AISC Code Deflection – Section L3 Deflections under service load combinations shall not impair the serviceability of the structure Limits on deflections are specified in building codes NYS Building Code Section 1604.3 Floor members Live load deflection ≤ span / 360 Dead load plus live load deflection ≤ span / 360 ACI Masonry Code Beams supporting masonry Total deflection ≤ span / 600 ≤ 0.3 inches Very stringent criteria

Deflections Beam deflections for various load conditions shown in table 3-23 For a simply supported beam with a uniform load on its full span Δ = 5wL4 / 384EI Use consistent units, usually kips and inches If load, w, and span, L, are known this equation can be solved for the moment of inertia, I, that would produce a given deflection For Δ ≤ span / 360, I ≥ wL3 / 43 For Δ ≤ span / 240, I ≥ wL3 / 64.5 For Δ ≤ span / 600, I ≥ wL3 / 25.8 In these equations, w is in kips / ft, L in feet, and I in in4 Approximate deflection in beams with non-uniform loads Calculate the equivalent uniform load based on moment E.U.L = 8 M / L2 Use this value for w in the equations above