Compression Members and Buckling
The design of steel compression members is governed by the tendency of slender elements to fail by buckling at stresses below the material yield strength. The critical buckling load for a perfectly straight, elastic column was derived by Leonhard Euler in 1757 and remains the fundamental basis for column design today. Euler’s formula states that the critical buckling load is proportional to the column stiffness and inversely proportional to the square of the effective length. The effective length accounts for the end restraint conditions, with a pinned-pinned column having an effective length factor of 1.0, a fixed-fixed column having 0.5, and a cantilever column having 2.0. The slenderness ratio, defined as the effective length divided by the least radius of gyration of the cross-section, determines whether a column will fail by elastic buckling or by inelastic buckling at stresses above the proportional limit.
The AISC Specification provides column design curves that account for residual stresses, initial imperfections, and inelastic behavior that reduce the capacity of real columns below the ideal Euler curve. The residual stresses locked into hot-rolled steel shapes during the cooling process cause portions of the cross-section to yield at applied stresses below the nominal yield stress. The effect of residual stresses is most significant for intermediate slenderness ratios where the column behavior transitions from inelastic to elastic buckling. The column curves in AISC 360 are based on statistical analysis of hundreds of column tests performed over several decades, providing design strengths that ensure a consistent level of reliability across the full range of slenderness values. The nominal compressive strength is calculated using the critical stress from the applicable column curve multiplied by the gross cross-sectional area, with appropriate resistance factors applied.
Local buckling of the individual plate elements that make up a column cross-section must be considered in design to prevent failure before the column reaches its overall buckling capacity. The width-thickness ratio of flange and web elements determines whether the section is compact, non-compact, or slender. Compact sections can develop their full plastic moment capacity before local buckling occurs. Non-compact sections reach yield stress but not the full plastic moment before local buckling initiates. Slender sections experience elastic local buckling at stresses below yield and require reduced design strength using the effective width concept. The AISC Specification provides limiting width-thickness ratios for each classification and prescribes the appropriate design approach for each category.
Flexural Members and Lateral Buckling
The design of steel beams must consider both the flexural strength at the cross-section level and the stability of the member as a whole. The nominal flexural strength of a compact steel section is equal to the plastic moment capacity, which is the yield stress multiplied by the plastic section modulus. The plastic section modulus accounts for the full yielding of the cross-section and is approximately 10 to 20 percent higher than the elastic section modulus for typical wide-flange shapes. Non-compact sections reach the yield stress but may not achieve the full plastic moment before local buckling of the flange or web limits the capacity. column slenderness ratio and euler buckling formula. lateral torsional buckling of steel beams. bolted connection design for steel structures. Slender sections are limited by elastic local buckling and use reduced effective section properties to calculate the nominal moment capacity.
Lateral-torsional buckling is a failure mode unique to beams where the compression flange buckles laterally and the cross-section twists. The critical moment for LTB depends on the unbraced length, the section properties, and the loading conditions. Beams with the compression flange laterally braced at close intervals can develop their full plastic moment capacity. Beams with intermediate unbraced lengths fail by inelastic LTB at stresses between the yield stress and the elastic buckling stress. Beams with long unbraced lengths fail by elastic LTB at stresses well below yield. The AISC Specification provides design equations that cover the full range of LTB behavior from fully braced to elastic buckling, with the nominal moment capacity varying as a function of the unbraced length.
Shear design of steel beams must ensure that the web has adequate capacity to resist the shear forces from applied loads. The nominal shear strength depends on the web area and the web slenderness ratio. Stocky webs with low height-to-thickness ratios can develop their full plastic shear capacity before buckling. Slender webs may experience shear buckling at stresses below yield, requiring reduced design shear strength using post-buckling strength or tension field action in certain cases. The AISC Specification provides equations for shear strength that account for the web slenderness and the presence of transverse stiffeners that improve post-buckling shear resistance.
Connections in Steel Structures
Bolted connections transfer forces between steel members through shear in the bolts and bearing between the bolt shank and the connected material. High-strength bolts in A325 and A490 grades are the standard for structural connections, with diameters ranging from 1/2 to 1-1/2 inches. Bolted connections are classified as bearing-type connections where the bolts bear against the connected plies, or slip-critical connections where the clamping force transfers load through friction. Slip-critical connections are required in connections subject to load reversal, fatigue, or where slippage would cause unacceptable deformations. The connection capacity depends on the bolt grade, diameter, number of bolts, bolt spacing, edge distances, and the thickness of the connected material.
Welded connections use electric arc welding to fuse steel members together without mechanical fasteners. Complete joint penetration groove welds develop the full strength of the base metal and are used for primary moment connections in seismic frames. The weld quality is verified through non-destructive testing such as ultrasonic testing, radiographic testing, or magnetic particle inspection. Fillet welds are used for simpler connections where the loads are lower and are designed based on the effective throat thickness and weld length. The minimum fillet weld size depends on the thickness of the connected parts, and the maximum size is limited to prevent excessive distortion and residual stress.
Connection design must consider the forces, the available space for installation, and the constructability of the connection. Eccentric connections that introduce bending moments in the bolts or welds must account for the additional stresses from the eccentricity. End plate connections and tee-stub connections are common moment connection types that must be designed to develop the required moment capacity. The connection stiffness affects the overall frame behavior, with simple connections assumed to rotate freely and moment connections assumed to provide full rotational restraint. Partially restrained connections with intermediate stiffness require more sophisticated analysis to determine the actual distribution of forces.
Tension Members and Hangers
Tension members are structural elements subjected to axial tensile forces that must be resisted by the net cross-sectional area after deducting holes for bolts or other connections. The design tensile strength is the lesser of the yield strength on the gross section and the fracture strength on the net section. The limit state of yielding on the gross section ensures that the member does not undergo excessive elongation under service loads. The limit state of fracture on the net section prevents sudden failure at the connection. The net section efficiency is reduced by the shear lag effect, where the tensile force is not uniformly distributed across all elements of the cross-section because some elements are not directly connected. The shear lag factor accounts for this reduction and depends on the connection geometry and the member cross-section.
The design of tension members also must consider the slenderness limitations that prevent excessive vibration and sag under self-weight. The AISC Specification recommends that tension members have a slenderness ratio not exceeding 300 for main members and 400 for secondary members. Pin-connected tension members require special design considerations to prevent failure at the pin hole through the limit states of tear-out, bearing, and net section fracture. The pin itself must be designed for combined bending and shear. Cables and rods used as tension members have unique design requirements related to the end connections, stress concentrations at fittings, and the potential for corrosion at the anchorages.